I would suggest spending a few extra dollars and going for a 20gal long (it's shorter and wider than a standard 20gal tank). It's still going to be less than ideal for space, but it'll do for a bit.
1) a good filter, which someone else can recommend because I've had no luck with internal filters.
2) a basking dock (I like the Zoo Med turtle dock)
3) Heat lamp and uvb bulb, or a combination uva/uvb lamp
4) NO GRAVEL! If you'd like to use something on the bottom of the tank, buy some river rocks at the dollar store
5) Two thermometers, one for the water, one for the basking spot
6) possibly a water heater, depending on how warm it is in your house
7) Plants for them to rest on
8 ) Water conditioner, anything suitable for fish is fine. That way, you don't have to let the water sit out for any length of time.
I'm sure I'm forgetting something.....but that's a good start.
I just realized you'd said you already have a 10gal. Make upgrading a priority, saving whatever spare cash you can
Good luck

The things that come to those who wait may be the things that were left by those who got there first - Steven Tyler