Yes, you can play most ds games on a 3ds! There are a couple exceptions however, like some games bought outside your region (the ds is region free, however the 3ds is region locked) and a couple other weird situations. I only own one ds game that I have to play on my ds lite though, so it probably won't be an issue.
There are multiple versions of the 3ds, and each with a normal and XL variant, depending on what size screen and console you prefer. I'm not sure how you are buying it, and therefore what versions you're more likely to encounter, but generally:
- The original ones are cheaper, but some people had problems with the hinge snapping if they were too rough with it, and it does have worse hardware.
- the new 3ds is more expensive, but has better hardware (so will run games a bit better), and has an extra c-stick button, which helps control some games. (it's also better for modding I think, if that's up your alley).
-there's also the 2ds; not many people's preference, but some people prefer how it feels to hold.
But generally, for most games, either will be perfectly fine. I still have my original 3ds, and I have not encountered any issues, other than slightly more lag in a couple games, but for me at least, its not that noticeable.
Sorry if that was a bit long-winded
