As many have said, underexposing and then boosting in post tends to show noise in the shadows. Two things I do when there isn't much light. One: I use the power of digital and, at really slow shutter speeds with static subjects, I shoot multiple shots of the same thing. I am not talking about burst mode, but, four, five or six identical seperate shots. On the computer one of those is almost always way sharper than the rest. One shot or six shots, the cost is the same. Two: Sometimes if I boost the shadows and they start to look bad, I decide if I really need the detail there. If I don't I will let them stay dark, hiding the noise, and then do something to draw the viewers eye to the main subject. I would usually rather have dark shadows than noisy shadows.
...Cam

I agree with a lot of the points already made here....

To minimize noise in images, do all you can to avoid under-exposure. If you have to boost exposure in post-processing, that will always amplify noise. Pf course, it helps to minimize cropping, too.

It seems counter-intuitive, but often works out better to use a higher ISO that allows slight over-exposure.... not to the point of blown highlights. However, when there are light bulbs in an image (as in this case), or exterior daylight coming in a window, you WILL see the histogram going off the right edge and shouldn't be too concerned about it. Higher shutter speed and smaller aperture might be useful, too... depending upon the lens and your skills hand holding shots.

BTW, there are other types of supports besides tripods.... a monopod or even a "poor man's tripod". That's a length of cord with a bolt on one end that screws into the tripod socket of your camera and a loop on the other end that you step into, to hold the cord taut while taking a shot. Believe it or not, it works! Steadies your shot.... plus you can coil it up and tuck it in a pocket.

I don't find the noise in the original poster's images objectionable. In fact, it sort of ties into the vintage theme of the room that was photographed. And it's really not obvious unless you look at the image really large... 50% or larger (... and "50%" is about 20x30" print equivalent, on most computer monitors these days, with an uncropped image from a 24MP camera).

However, previous posts are correct that various noise reduction methods. In my opinion, it's most ideal to do NR from a RAW file... And NR should be done early in post-processing, before changing size and DEFINITELY well before doing any sharpening (which is usually best done as one of the final steps).

I hope you don't mind, I tried Imagenomic Noiseware on your photo, just to see what it could do. (I have Noiseware as a Photoshop plug-in, but also available as a stand alone software.) I only applied enough to reduce noise, tried to maintain as much detail as possible, and backed off when artifacts started to appear. If it were my image, I might lighten it further and experiment with some more advanced techniques in Noiseware. Here there's still some noise, but it's reduced a bit:

Thanks so much for your response - this is great information to have and definitely some things I can try. I don't typically shoot indoors, but it's a skill I'd like to acquire. I've seen several other replies here about upping the ISO even though you're right, it does seem counter-intuitive. I'm going to be shooting my great, great niece and nephew (1 1/2 and 3 years old) on Christmas Day opening their packages, so will absolutely need a higher shutter speed for that! I'll take your suggestions and practice a bit at home before the big day to see what I can get. And thanks for trying your modifications - always good to learn about other software tools that are available out there.

Just got back from a wonderful trip to see the Biltmore Estate Christmas decorations and when I started reviewing my photographs, I noticed that there seemed to be a lot more noise than I would have expected at ISO 2200. I've heard all the great things about how the D750 can go up to 6400 and the noise is "acceptable" but this doesn't look right to me. So I'd like your opinions - maybe there's a setting I don't have correct? Or am I just overthinking it? Most of the photos I took were inside, and we weren't allowed to use flash or a tripod, of course, so these were all handheld, Manual mode, Aperture set at 2.8, shutter speed anywhere from 1/30 to 1/60, and ISO up to 2200. I'm including one of the photos for you to take a look at. Caveat: I did bump up the exposure in this photo bef posting as it was quite dark. If you'd like the original, I can re-post. Any opinions / feedback / recommendations / etc. will be appreciated!