Some general rules of thumb regarding teleconverters...

#1. They work best on prime lenses, less well on zooms.
#2. There is always some loss of image quality with any TC, so they work best with extremely high quality lenses that can tolerate some loss of IQ.
#3. With few exceptions, a more powerful telephoto lens without a TC will out-perform a less powerful telephoto with a TC.
#4. A weaker 1.4X or 1.5X teleconverter will "cost" less loss of image quality than a stronger 1.7X or 2X. Even stronger 3X are largely worthless.
#5. There is ALWAYS light lost to a teleconverter, which will slow autofocus a little to a lot... or cause it to completely fail. A 1.4X "costs" one stop. A 2X costs two stops worth of light.

A Tamron 70-300mm f/4-5.6 lens will "become" a 98-420mm f/6.3-8 with a 1.4X attached. Or, it would "become"a 140-600mm f/9-11 when combined with a 2X TC. You didn't indicate what camera you will be using this on, and need to check if its autofocus system is able to work with slower than f/5.6 or less than f/8 max aperture. Many cameras will not be able to AF the lens + 1.4X and likely none would be able to do so with the 2X, due to insufficient light.

It also would help to know what camera, because there are different possibilities. For example, Canon's 1.4X III and 2X III are very high quality and cost about $430 apiece, but have a protruding front element that needs to fit inside the rear barrel of any lens they're use upon.... and most 70-300mm cannot accommodate them. I don't know whether or not they can be used with the Tamron. Nikon's 1.4X III and 2X III are also excellent, cost almost $500 each, but are recommended for use only on select lenses (consult the Nikon website). I have no idea what's offered for use with Sony, Pentax, Olympus, etc.

There are third party teleconverters, too... though with all the possible combinations, it's hard to predict what will work well and what won't.

Kenko offers two 1.4X models that are affordable and have a good reputation.. But which would be best to try once again depends upon the camera it will be used upon (as well as the type of photography being done, to some extent). The Kenko 1.4X "MC-4" is quite sharp in the center with most telephoto lenses and it only costs around $110. It is not quite as sharp in the corners or at the edges, but for many types of photography that might not be a concern. It also may be a good choice for use with an APS-C sensor camera that's only going to utilize the central portion of the image anyway. A "full frame" camera user might prefer the Kenko 1.4X "Pro 300" model that's a little more expensive at about $140, but has better sharpness in the corners and at the edges. Kenko has recently introduced an "HD" version of their teleconverters for some lens mounts. I don't know how their image quality compares and they tend to be a bit more expensive. I also don't know how their 2X compare.

Some lenses simply work well with certain teleconverters, while others don't. And it's a bit subjective.... results that you find acceptable, I might not.... or vice versa. It also depends upon what you want to do with the images.... if you just post them online at typical internet sizes and resolutions, you may not notice the loss of IQ due to the teleconverter. But if you make large prints from your images, you're likely to be less tolerant of IQ losses.

I have Canon 1.4X II that I frequently use with 500mm f/4, 300mm f/2.8 and 300mm f/4.... and more rarely with 135mm f/2, 70-200mm f/2.8 and 70-200mm f/4. It's said to work well with a 100-400mm I recently got, too... but I haven't really tested it much yet.

I also have Canon 2X II... but pretty much limit using it with 300mm f/2.8... occasionally with 500mm f/4 though unable to autofocus on some of my cameras.

Ultimately....

If money is not a concern, you'd almost be better buying a more powerful lens such as a 100-400mm or 150-600mm or 200-500mm (depending upon what's available for your particular camera). This will be larger and heavier than what you have now... enough so that there's good possibility you'll also want a good stable tripod or at least a monopod to use with it.

But for travel and