There are multiple devices that play DS games
1) DS
2) DS lite
3) DSi
4) DSi XL
5) 3ds
6) 2ds
7) 3ds xl or 3ds ll depending upon the region.
8) n3ds or new 3ds.
9) new 3ds xl/new 3ds ll in Japan.
10) New Nintendo 2DS XL
I might also note 1a) There are later batches of original DS that with a custom firmware can gain the variable brightness of the DS lite which is quite desirable, screen perhaps not quite at the level of DS lite but still better than stock. No cases of "these have better buttons" or fancier screens (think AGS 101 for the GBA SP) or anything like that.
Some of those also got Chinese (usually going under the name ique) and South Korean iterations of the hardware. In some cases there are some technical differences, and might need some hacks to get them working properly, but nothing in terms of screen, form factor, button type or so forth.
There are no hardware mods to speak of like we are seeing with the original GBA and backlights on the screen, there is an overclock thing but eh really. Similarly I don't think we have seen clone consoles yet either. There are no ways of officially playing the DS on a TV a la the super gameboy (
https://loveconquersallgam.es/post/2350461718/fuck-the-super-game-boy-introduction ), game transfer pack or gb player (
https://www.gc-forever.com/wiki/index.php?title=Game_Boy_Interface ).
As you are on GBAtemp I assume you care about modded games and running them your own way. For the most part I suggest a flash cart to do it with. The DSi and 3ds have the option these days to run them in software -- it was not available for a long time, and still has limitations such that I will call flash carts the superior method, but with a bit of fiddling you can enjoy vast swathes of the DS library.
Technically there are DSi exclusive games and games enhanced by DSi hardware. Practically though nobody cares, and the few bits of nominally interesting DSiware were backported to the DS.
http://www.rfgeneration.com/blogs/pdrydia/DSi-enhanced-titles-an-incomplete-list-1969.php
Few will say the screen for any of the 3ds/2ds family is the superior method for DS games, more that it is passable at best (though probably still rated ahead of the original DS by many). If you also want 3ds games though then you can go see what goes there. Some of those models are Japan only but hacks can break them out of region locking (region locking mostly appeared on the DSi and carried on into the 3ds). Also unlike the DSi then the new 3ds does have some exclusives and enhanced titles for it --
https://www.neogaf.com/threads/games-that-are-exclusive-enhanced-on-the-new-3ds.1177455/
Personally I find the 3ds library massively underwhelming when compared to the GBA and DS (both of which I loved), and if you had played the DS library then there is even less on the 3ds for you as most things were weak refinements of DS games, indeed I can't really think of anything that you would want to go find a 3DS entry in the series for rather than play the DS version, I can think of several things I would go in for the DS version for over the GBA.
Few will argue the original DS screen to be the best. I like mine but I am apparently a freak so eh. The form factor is also pretty agreeable to my hands where the later iterations have something lacking. Speaking of form factor I have not yet seen any replacement cases or covers go widespread to make things more like a controller like we have seen with some of the older devices.
The DS lite for many is the superior method of playing DS games.
Some like the DSi screen and would favour it over the DS lite.
Flash carts then. Arguably the best DS flash cart is the supercard DSTwo and DSTwo+ (the + iteration appearing later in the 3ds lifetime and acting as a second run of DSTwos which were getting rare, the plus itself is reasonably hard to come by nowadays. The plus is occasionally also known as the supercard 4 in 1). Unlike other carts, certainly all those that will be in the final list, the DSTwo comes with extra hardware that homebrew can use. For most practical purposes this means a decent video player that you don't have to convert for, a GBA emulator on a DS cart (the DS slot is too slow and the DS has too little memory to properly handle GBA games so they need to be handled onboard, more on that in a bit), a bit better DOS emulation, and better SNES emulation than a vanilla DS. This extra hardware means extra cost, and a bit more power drain. There are no issues getting it working on a stock DSi or 3ds either -- some other carts will need the DSi/3ds to be modded and its protections (
https://hackmii.com/2010/02/lawsuit-coming-in-3-2-1/ ) turned off.
For the rest we were having a nice little discussion
https://gbatemp.net/threads/current-favored-kit-for-ds-dslite.537988/ the other day. The short version here is if you are purely in it for the DS/DS lite then some older cards are good stuff, a hacked 3ds also affords you some more options. For a stock dsi or 3ds then you are probably going to have to wade into the world of things named R4*.
*the R4 was a factory made version of the M3 (M3 had made a name for themselves in the late GBA and early DS era). Unlike some other things it supported drag and drop from the start, I did not think this ideal but it took off in a big way and thus the parents/clueless person term for DS flash cart became R4 (think how some people will call any hack a jailbreak, or maybe root these days). By the time of the DSi/3ds then basically everything that was not a company tracing its way back to the early days of the DS or even GBA seemed to have R4 in the name somewhere, some were actually not bad though if you only wanted games (including modded versions thereof) to run, and maybe have a few cheats on said same.
GBA stuff. As mentioned the DS slot is too slow to handle GBA games so if you want to play GBA games on something with a GBA slot (so the DS or DS lite)you will want a GBA slot flash cart if the DSTwo is not enough for you (it is good but not flawless, it also goes toe to toe and quite possibly wins with any other method made for the DSi and 3ds to play GBA games -- DSi and 3ds homebrew is pretty weak compared to the GBA and DS, mainly as by this point android and such was pulling all the focus). 4 main contenders here.
1) EZFlash 3 in 1
2) EZFlash 4 aka EZ4
3) EZflash Omega
4) Everdrive X5
The 3 in 1 is cheaper and requires a DS flash cart to manage it. Good luck finding a full size GBA slot version today -- the ones made for the smaller GBA slot in the DS lite are what you will find.
The EZ4 is a bit older, can probably be found for a bit cheaper and does well. For many years it was the main choice for GBA flash cart, or at least the only one people could find and actually did very well.
The Omega was EZFlash's answer to everdrive. It has all that the EZ4 had in the end plus real time clock (useful for some pokemon titles) and a few features not seen on the EZ4 like savestates (savestates had been a thing for the EZ3 but never made the leap to the 4, note the EZ3 is not the same as the 3 in 1 above)
Everdrive made a bit of a habit of rolling into flash cart scenes, making a flash cart using modern tech and then becoming the clear and undisputed best cart there. This was their attempt for the GBA and it is very nice, bit expensive and sticks out a bit from even a normal size GBA slot. On the other hand even when the GBA was current we had devices that would house a sizeable fraction of its library and play just about everything (
http://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995 ) just like they were originals.