I agree with some of the recommendations already given... But I disagree with some others.
Yes, you might be wise to work with what you've got already and get to know it, before buying anything different. And, yes, you can shoot landscape photos with the lenses you have. That 18-55mm is moderately wide angle, as well as zooming to normal and short telephoto focal lengths.
I don't agree with a suggestion to consider the Sigma 17-70mm lens. That's virtually the same as you've already got in your 18-55mm, so really wouldn't expand your capabilities at all. About the only thing you'd gain is a somewhat larger aperture, and that's not particularly useful for landscape photography where most of us are stopping down to smaller apertures most of the time anyway.
I also don't agree that even wider lenses should be ignored, as some earlier responses appeared to suggest. Yes, you can use other lenses. But many landscape photographers like to use wide angle lenses... wider than what you've got now. So that's something you might consider in the future. This is not to say that less wide lenses and even telephotos can't be used for landscape photography. They certainly can and you're pretty well set up to do so already. But it's still quite arguable that landscapers like their wider lenses... a lot!
Unfortunately, one of the most affordable wide angle zooms isn't usable on your D5100. The Nikkor 10-20mm DX VR is small, light, inexpensive and gives an ultrawide angle of view (which most define as 15mm or wider, when used on an APS-C camera like yours). At around $300 new, that Nikkor is one of the cheapest. In spite of the affordability, it also has quite decent image quality and even offers VR image stabilization (which is rare on ultrawides). HOWEVER, that's a relatively new "AF-P" lens, using a stepper motor to drive the autofocus. The D5100 is incompatible with that type of focus drive. For autofocus you need AF-S Nikon lenses, or similar from third party manufacturers.
This means if (or when) you want wider, you'll need to consider lenses other than that AF-P Nikkor. Nikon themselves offers two AF-S DX ultrawides: 10-24mm and 12-24mm. But they are ridiculously expensive at around $900 and $1100, respectively. So you will probably end up looking at third party lenses. Tokina, Sigma and Tamron offer various options that are able to autofocus on your camera.
Tokina has offered 11-16mm f/2.8, 11-20mm f/2.8, 12-24mm f/4 and 12-28mm f/4. All these are pretty darned good lenses, well made and in some ways they even emulate Nikon lenses (they use the same "DX" designation, plus their zoom and focus rings rotate the same direction as Nikon's... which are opposite most other manufacturers). The f/2.8 Tokina were the largest aperture available in ultrawides, but that tends to make for larger, heavier lenses and some increased susceptibility to flare. The original Tokina 11-16mm and 12-24mm in Nikon mount also do not have built in focus motors, so would be manual focus only on your D5100. The "II" version of those lenses, as well as the current models, acquired focus motors in the Nikon versions, so will be able to autofocus on a D5100. The 11-16mm also trade off that very narrow range of focal lengths, in order to have an f/2.8 aperture. They're sharp, too, but among the worst for flare.
For most landscape photography, f/2.8 is unlikely to be needed. While folks shooting astrophotography might appreciate it for the brighter viewfinder at night, most daytime landscape photography is done using smaller lens apertures, making it sort of a waste. I've used the Tokina 12-24mm f/4 lens myself and found it good (though a Canon 10-22mm I use now is better... sharper corner to corner, less chromatic aberration and among the most resistant to flare).
Sigma currently offers a 10-20mm f/3.5 HSM (see image examples posted in an earlier response), which used to be one of the most expensive ultrawides, but has been steeply discounted and more affordable recently. At around $650 new, it's still more than you wanted to spend. It's also a rather large, heavy lens. In the past Sigma also offered a smaller, lighter, less expensive 10-20mm f/4.5-5.6 HSM. Again, do you need any larger apertures? That lens was discontinued a couple years ago, but can still be found used at favorable prices. Sigma also offers an even more extreme 8-16mm, but that's crazy wide and has tons of perspective exaggeration. It's not a fisheye lens, but close to it! Finally, Sigma also offers a 12-24mm (now in its second or third version), but it's a full frame lens... very large, quite expensive and has a convex front element (same with the 8-16mm) makes it impossible to use standard screw-in filters.
Tamron started it all by offering possibly the first ultrawide lens for APS-C cameras with their 11-18mm lens many years ago. Honestly, that lens was awful. It had pretty poor image quality. The did a lot better with their 10-24mm offered a few years later, and have improved it still more with the latest version that adds VC image stabilization.
I mention image stabilization (VR on Nikon, VC on Tamron, OS on Sigma although none of the above Sigmas have it) because that might be helpful to you with some difficulty holding a steady shot. If you use a tripod that's even better, but some lenses you should turn off image stabilization when putting it on a tripod. Check for details and advice about this for any stabilized lenses you use (including the three you got with the D5100, which all have VR). It's rare to find image stabilization on ultrawide lenses, simply because it's not all that necessary. Wide angle lenses are relatively easily hand held even at slower shutter speeds. Image stabilization is most helpful on telephotos.
Another response mentions third party, manual focus lenses, which can be quite affordable. Not long ago I bought a Meike 12mm f/2.8 lens for a mirrorless camera system for the princely sum of $120! It was a demo unit, but normally sells new for under $200 (it also sells as an Opteka lens). I wasn't expecting much, but was pleasantly surprised at the build quality and image quality it produces! See example image below. While that particular lens isn't available for Nikon F-mount that your D5100 uses, there are other manual focus (and manual aperture) lenses available. Rokinon/Samyang, IRIX, Venus Laowa and some other brands offer them. Two things to consider... since you already have 18mm wide, I'd probably look for 15mm, 14mm or wider. Some of those are fisheye, which have strong curvature effects in images. Also, some of these lenses have a convex front element, precluding use of standard filter. Other than these possible concerns, they may be well worth consideration. Most landscape photography can be done easily with manual focus and even manual lens aperture control (lens will actually stop down, dimming your viewfinder... If that's a problem at small apertures, instead try using your camera's Live View on the rear LCD screen for composition and focusing).
Several have commented that your plan to get a 40mm macro lens might be a disappointment because at high magnifications that short focal length will make for very little working distance between the front of the lens and your subject, and that can be a problem. It might scare away shy live subjects... or cause you to cast a shadow over inanimate ones. I agree, that's definitely something to think about. A macro lens in the 85, 90, 100 or 105mm range is generally the most broadly useful. Longer than that render very shallow depth of field and get more difficult to hold steady at high magnifications. There are some slightly shorter... 70mm, 60mm and even 50mm... that might be more compact and can serve well, too, especially if not often using full 1:1 magnification These lenses can be fine for flowers, for example... which are more often shot at 1:2 or 1:3 or less magnification. These shorter focal lengths also can serve well in studio close-up work, where you have more full control over lighting. I often use a 45mm lens for small product photography.
Before taking the plunge for a macro lens, you might just try using macro extension tubes with the lenses you already have. A high quality set of Kenko tubes (12mm, 20mm and 36mm) sells for about $130. There are less expensive from Opteka, Zeikos, Fotodiox, Vello and others... $60 to $75 a set. There are even "economy" versions of some of those, more plasticky but as cheap as $50 or so. All these have electronic contacts to support full lens functionality (the Nikon brand extension tubes DO NOT, even though they are much more expensive). Avoid the really cheap macro tubes, costing $25 or less. Those lack any electronics, so are really only usable with vintage lenses.
However, it also bears mentioning that there are many vintage macro lenses that are excellent and can be bought quite cheaply used. I picked up a Tamron SP 90mm f/2.8 "Adaptall" lens from a local secondhand store for all of $20. The Adaptall system allows the mount to be changed, though this one came with a Nikon F-mount on it. I bought a replacement from China so I could use it on Canon EOS DSLRs. Even though Tamron discontinued Adaptall lenses and their adapters many years ago, they are still being made and even to fit camera mounts that didn't exist when the lenses were originally made. More recently I swapped the lens mount yet again to be able to use it on a mirrorless camera.
There are many other excellent vintage macro lenses. Nikon themselves made some of the best. But basically anything with Nikon F-mount (or an interchangeable mount) might be worth consideration. Manual focus and manual aperture lenses will be slower to work with, which can be challenging with faster moving subjects. Using the manual Tamron 90mm lens, it me about 75 shots to get a few I liked and were sharply focused, of this bee flying from flower to flower seeking nectar...
Finally, someone mentioned the sky was being over-exposed in their scenic shots. That's often a problem for all of us. The sky is brighter than it looks and if you make an exposure for the ground and the rest of the scene before you, not adjusted for the sky, it will often end up over-exposed. If you adjust exposure so the sky is right, then the rest of the scene ends up underexposed. Fortunately there are some relatively easy solutions.
Back in the days of film we carried graduated neutral density filters.... one stop, two stop, three stop... which were half clear and half tinted gray. Those were used to "hold back" the sky. But they were bulky, good ones ain't cheap and they were never ideal because the horizon line is rarely straight and a perfect match for the transition line in the filter. As a result, the filter overlapped part of the scene.
Today we have better options with digital. Often it's possible to just make two exposures... one for the sky, the other for everything else, and then combine the "correct" parts from each later in post processing. This doesn't work with moving subjects, but is fine with many landscape photos. Another solution is to shoot RAW... which allows for more adjustment... and then double process the image, adjusting one for the sky, the other for the rest of the scene, and then combine the "correct" part from each into a finished image. This works with moving subjects. Either of these multi-image techniques is far more controllable and precise than using graduated ND filters like we did in the past.
A third solution is to simply install a circular polarizing filter. That will deepen the blue of the sky in many cases. Polarizing filters work best when the light source is at 90 degrees, such as sun overhead when shooting a scene or way off to one side or the other. Polarizers have least effect at 180 degrees or 0 degrees... such as light source directly opposite the direction you're shooting... or when shooting directly at the light source (so take the filter off for sunset/sunrise shots). The shots below were done using a polarizing filter...
You do have to be careful of one thing. Because the effect of a polarizer is uneven... it tapers off closer to light sources... and because a wide angle lens captures a broad expanse of sky... you have to decide if this is a concern. The first shot below shows how this effects an image where the light source is to the left and the sky is very clear. I saw this in the viewfinder and decided to shoot anyway because I didn't mind the effect for this particular image. The second shot is similar, but the uneven effect of the filter is largely hidden by the lighting and in cloud details in the sky.
One other type of filter is popular for landscape photography... standard neutral density are used to allow longer exposures, such as when wanting to make flowing water appear blurred. If you shoot around water a lot, you also may want a plain UV filter or clear filter to protect your lens from any spray, too.
If you plan to use filters, some of the ultrawide lenses that have convex front elements can be a problem. They won't allow standard screw-in filters, so you may need to buy a pricey, special filter holder and oversize filters to use on those lenses. It's something to consider, when shopping.
Enjoy your new kit... and don't be too quick to buy more... unless you really, really want it!