Short rundown of the available methods. Again
http://gbatemp.net/threads/how-to-dump-your-own-3ds-ds-games-saves.274290/ has more for some of them.
Ancient.
Old GBA slot flash carts with SRAM. Run some DS homebrew off this and it will dump it to the GBA cart SRAM and allow you to assemble it back together later.
Don't do this, I mention it mainly so if you go searching and see something you don't spend too much time contemplating it.
GBA slot with DS homebrew and DLDI.
Originally the EZ4 was the cart of choice for many here as it was readily available but the newer loaders and new batches/models do not have this.
Rudolph's dumping tools are the ones of choice for most people here.
DS slot flash cart with EZ 3 in 1.
The protocols to handle the GBA slot EZ 3 in 1 are known so you can run a DS homebrew program from a DS flash cart, eject the flash cart and insert a normal game before dumping it.
You will need a DS slot cart for this. Technically you could have another means to launch homebrew and then write to it but you won't have such means. Any DS slot flash cart will probably do here so get one of the older R4i things that does not work on current dsi/3ds models as they go cheap, or buy a good DS flash cart.
DS/DS lite with flash cart and wifi.
In this you run a DS program from anything you have that runs it. Insert your game into the DS slot and it sends the ROM/save to a FTP server or in some cases netcat.
Other than the ancient stuff there are two main groups of programs people use
1) Rudolph's dumpers. These have more polished GUIs and do saves more readily.
2) Wood dumpers. These dump newer ROMs more accurately, though nobody cares as the thing that is missed with Rudolph's is just a pointless part of the header (
https://hackmii.com/2010/02/lawsuit-coming-in-3-2-1/ covers the difference with newer games). Not sure what goes right now for saves or some of the other methods.
Above it would now seem that the 3ds, by virtue of having a standalone custom firmware*, has the ability to grab saves and the like.
*technically there was a custom firmware for the DS with some measure of cart bothering but it is not worth it when you have flash carts instead.
Finally there are PC based devices to handle it, I suppose there are also action replays but nobody cares about those.
Anyway the NDS backup Adapter family is the more common one. The plus model doing the pokemon gold and silver remakes (the pokewalker thing they came with was unique in DS game carts and abused the save pins to have IR communications), you can mod a cart to bypass this and there are also special versions of dumping tools to handle it for some of the other means.
I don't know what means you have right now if you are familiar with soldering iron. It is mainly SPI (a standard chip to chip communication protocol) so it should not require anything too exotic
http://problemkaputt.de/gbatek.htm#dscartridgeioports