Security was blown wide open on the DS many many years ago and its firmware is minimal (though there are some hacks available, DSi being a step up of sorts had a proper firmware and onboard SD card that affords more options here).
To that end yeah flash carts are what you want. The DS slot is too slow to reasonably run GBA games so you will either need a so called enhanced flash cart there (which for most means Supercard DSTwo) or a GBA slot/slot 2 cart. The DSi again has a few more options here, 3ds even more so, but neither have a GBA slot.
DS slot stuff.
The DSi (and then 3ds) saw a series of updates which blocked DS flash carts, but at the same time there were still DS games coming out and many of those blocked would not see updates to run the very last DS games (see pokemon conquest and most things after that that, though there is a database to attempt to sort things manually where necessary).
To that end most things people look at today were either the few to make it through or things made later.
The Supercard DSTwo being the main thing to make it though. It is a bit power hungry but having an onboard CPU means GBA emulation, SNES emulation way better than what the stock DS and every other normal flash cart affords and a bunch of other things. Also has savestates, onboard cheat search, in game guide reader and more besides. For whatever reason they stopped making them way before demand dropped so good luck finding one today, much less for a decent price. The Supercard DS one is also nice but lacks some of the fancier features, some clones did pop up a while back on the usual suspects
https://gbatemp.net/threads/dsone-clones-appear-to-be-available-for-purchase.586226/ but quality is a bit variable shall we say where originals were not bad.
Acekard2i also being noted in this. Not the best hardware but its unofficial loader called AKAIO is widely regarded as having the best support (they certainly cracked more games than anyone else/were used as a base for many other things) even if not the fanciest features (cheats and soft reset, albeit reference grade versions thereof).
Everything else means you join the world of R4 "clones". The original R4 was basically an OEM M3 (somewhat popular GBA flash cart maker) but owing to it being drag and drop (inferior in many ways to the savelists and other things) and reasonably cheap it became the byword for flash carts on the DS (and in some cases remains it to this day). You then got other flash cart teams making things with R4 in the name, any computing/marketing buzzword you care to think of slapped on and sold as a DS flash cart. Today most things bear little resemblance to the original R4. That said you will want one Wood compatible (related to AKAIO above and with much the same features/limitations -- though can you play essentially every game with cheats and soft reset? Yes and that is all most people want) and without a time bomb (flash cart is a unique thing in marketing in that you tend to buy once, possibly for not a lot, and get updates for years afterwards. Seeing this many of the fly by night makers of R4s included a check on the internal system data in their kernel that stops it from working if it is too far in the future, presumably to get them to buy another as "it is not working any more"). I don't know what we are suggesting here for R4 clones though.
Call this the short version. Everything after this is not fluff but not necessary to getting DS games up and running.
If you find an older cart that made it to the DSi blocks you could probably have some fun -- the EZ5i for instance other than the enhanced emulators having features that might go toe to toe with the dstwo and sometimes even win. Wood is also made to run on original R4 carts as well but don't get one of those as you will be limited to 2 gigs of memory (they don't support SDHC, much less SDXC which is 64 gigs and up but I don't think any DS stuff does that)
GBA slot stuff.
Assuming you can not find something in the DSTwo family you are going to want this (there is very limited proof of concept stuff available but not something you would really want to play on). You can also do this as an when time, funds and desire allow for it and just stick with DS stuff which has a lot of good games (including remakes of GBA games and sequels that do much the same) and a potentially far cheaper barrier to entry.
There are many ways to classify things but for today
Expansion Pack
Legacy DS era device
Modern GBA flash cart
There is also legacy GBA era flash cart but save those for the collectors really. This would include the so called fire cards (when they say 128Mbit or 256Mbit, aka 16 megabytes or 32 megabytes they mean it) you might still bump into on various sellers and auction sites.
Expansion pack for most means the EZFlash 3 in 1. Originally paired with the EZ5 family of flash carts (and still having great internal support on them) it had its access protocols made open source and thus was adopted by everybody into most things and if not there is various DS homebrew tools to do things instead.
Today you will probably only find the DS lite sized ones (GBA slots in DS lites being small/original size GBA carts stick out) as GBA original sized ones were only a relatively smaller run and wanted for both repros/people wanting to burn their ROM hacks to a cart and seemingly some electronics project for some people (the RAM or something was really well suited to that).
They are a bit clunky needing the DS side of things to be managed, batteries often want replacing and yeah but cheap and plays basically everything in the GBA library.
If by some miracle you trip and fall on a M3 game expansion pack (they also had a separate rumble version, the EZ 3 in 1 has rumble onboard which is part of the 3 things in 1) you can possibly get it to work (the main homebrew author and them have a falling out so support is less/needed hacked versions of things) and it would also do things for you.
I should note that these (and a few vintage GBA things and DS things, and to a lesser extent the ram pack that came with the opera browser) can be used to boost a few pieces of DS homebrew
https://wiki.gbatemp.net/wiki/3_in_1_Expansion_Pack_for_EZ-Flash_V#Third-Party or indeed might be required for some (most will be just fine though, and there is some really really good stuff in DS homebrew).
Legacy DS era device. Various ones might be found but for most it will mean the EZ4 family (later also EZ redux and reform) or M3 family if you are really lucky. Supercard GBA slot things are still found from time to time, and very cheap but unlike basically everything else on the list these things are not good so don't do it.
The EZ4 family saw a series of updates to use SDHC memory and onboard patching in more recent years as well, albeit it at the cost of its DS mode which had a few uses for some people like dumping DS games and saves even if the 3ds is by far the better option for that one. Cheats are tricky but for that we have stuff like GBAATM/GBAATM rebirth which will hardpatch cheats into ROMs
https://gbatemp.net/threads/gbaatm-rebirth-gba-auto-trainer-maker-a-new-hope.564321/ https://gbatemp.net/threads/gba-auto-trainer-maker-gbaatm.99334/
Be aware the 3 big makers of such things (M3 also had G6 and that is the big four of the GBA slot DS era that most care about today) had a lesser GBA slot cart that aimed to play DS games. For EZFlash it was the EZ4 lite compact, for M3 , supercards had the supercard rumble.
Modern GBA flash cart.
While the difference between GBA era flash cart and what can be done with the power of modern technology is not quite as stark as the NES, SNES and N64 to now that saw everdrive utterly dominate much of the PS1 era on back flash cart world then some still tried.
Everdrive have a few GBA flash carts which are very nice indeed.
EZFlash came back as well with the Omega and later Omega Definitive edition.
You will pay quite a lot for these but if you want the all singing, all dancing, bells and whistles white glove treatment in flash cart form then they are wonderful. At the same time I will be the first to say
https://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995 covers all the list of tricky ROMs on the GBA and how to fix them. Accordingly said older things that are not supercards will play everything as well if not better than a stock GBA and fairly minimal fuss if you want them to. This is in stark contrast to the vintage older consoles and homebrew flash carts that was the state of things for basically every old console before everdrive took to the stage a few years back.
Do also note that while GBA flash carts (and certainly DS but that is a different matter) can use emulators (GBA emulation is about 95%, DS slightly better still) to play GB/GBC games then outside of a very rare super vintage flash cart and even rarer accessory then GBA carts do not work as GB/GBC flash carts -- the GBA, GBP and GBA SP all featuring GBC hardware and GBA slots). You can get those too (everdrive again, EZFlash also have the EZ Jr) but play that as you will. The main difference for most is either the inability to do the stretch thing in a GBA, the few glitches there are, and inability to do link play* between GBA and GB/GBC hosts.
*marginally relevant at this point but amusing
Suppose I had better do custom firmwares.
The DS has a firmware (it is what you see when you get the menu to calibrate touch screen, launch download play, launch whatever slot, menu and pictochat) which could be written to. Accordingly some people did do hacks to it. The most popular by a massive margin (to the point the others are curios for those super nerds looking into things**) is the flashme line of such things.
https://www.gamebrew.org/wiki/FlashMe
There are basically two versions -- stealth and normal. Older ones might do more things for DS code GBA slot carts but nobody uses them in anger these days (I say having one I use a lot) as they stopped getting DS updates in late 2009 at very best.
Stealth left the health and safety warning screen intact, the other does not. They all disable downplay play restrictions (since worked around other ways if you really care, not that most people even cared before that), can autoboot flash carts in the GBA slot in DS mode (act as a passme basically) and otherwise lose various restrictions that might have been in place. Most then use it as a nominal backup (it has a means to boot a minimal firmware if the firmware gets overwritten by a piece of malware or something) and more importantly for losing the very annoying health and safety screen which sees you press a button every boot (though some flash carts can sort of bypass it a bit) presumably because some lawyer or health and safety type somewhere.
Installation is relatively easy. Get something that runs DS code (you only need it for a few minutes), run installer, short relevant part on the motherboard to enable firmware writing which you can usually do by opening battery cover (though if you want to take the back panel off then might make it easier in some ways), watch percentage bar crawl up, reboot. The shorting part is more tricky on the DS lite as you can easily bridge it to something nearby and it will turn off (possibly blow a fuse but turning off is the problem), which if flashme is not fully installed yet could be tricky (you need to get to 5% I think it was to get the failsafe installed). A carefully made tool will do it fine though where original DS could be done with tin foil or a screwdriver or whatever was conductive and fit in the hole hence the difference in the minds of some.
**for the sake of the forum searcher. Loopy's (same guy that did flashme, various homebrew things and later the jumbotron DS video capture and 3ds video capture) minimal firmware which is much as it sounds and nice black screen text job (albeit too minimal for most and I am normally all about the black screen-text loaders), creebome (
https://gbatemp.net/threads/creebome-ds-firmware-released.43280/ ) and fwnitro (probably the closest thing to a upgrade as it could theoretically apply some cheats to various things that might not have otherwise had the option, practical realities were different and cheats could even be made to work with original carts with DS homebrew like
http://www.chishm.com/NitroHax/index.html, and hardcoded into ROMs quite easily with stuff like DSATM.
