To play roms on your DS, you will (of course) need a special cart to copy them onto, as well as (for most modern flashcarts) a smaller, fingernail-sized bit of memory called a MicroSD card to copy games onto.
If you just want quick and cheap, though, the DSTT can be gotten for as low as $6... the Acekard is fancier in various ways depending on what version you get, but the DSTT is cheap and plays games fine. I have an M3real and it also works fine.
The price of MicroSD cards varies based on their capacity, ranging from as low as $4 or less for 1-2 GB cards, to ~$50 for 16 GB cards. Some devices will come with a small MicroSD card.
You copy the games onto the MicroSD card (along with some system files for your cart), then you put the MicroSD card in the cart and put it in your DS. And turn it on. It's as simple as that, and I probably made it sound more complicated than it really is.
And yes, roms will let you play online. I haven't heard of any copy-protection that prevents that.