You’re right in that you can’t use PS3 sticks on 360, Cheddar Jack. You can, however, mod your stick like Bruce_Deluxe said…but you may not be ready for that. I had this very same question and asked a bunch of high-level friends of mine, and they agreed with what’s already been said here: PS3 is usually used for big tournaments (EVO, etc.) because it’s so easy to swap sticks out – you just plug yours in, change the controller to whatever number you’re supposed to in the guide menu, and go. The 360 is really finicky in this regard so PS3 is obviously the best choice for tourneys where people will be rotating in and out on the same consoles all the time. Local, smaller events and casual play usually ends up on 360 just because that’s what people have. If you have a PS3, buy a PS3 stick and don’t worry about competition just yet unless you’ve got a high-level group you can go practice with that uses 360s. (You can probably use their sticks anyways if you’re doing casuals!) If you’re just now thinking about buying your first stick it’s gonna take you a while before you’re really ready to seriously consider competing, and by then you’ll probably be ready to mod your stick.
If you do want to mod your stick at some point to be dual or multi-console capable, check these threads out:
Official Cthulu/Chimp thread – Dual/Multiconsole custom-made PCB support and discussion thread
Paewang Revolution modding/selling thread – the Paewang Rev has a tiny, convenient PCB that’s 360 and PS3 compatible right out of the box, you can buy the bare PCB or the whole stick from laugh over there