10 gallons will only be sufficient for baby turtles. One rule of thumb I see is 15 gallons per inch of turtle length, though I think 10 gallons per inch could be a feasible minimum. This would mean that a full-grown red-eared slider (using this as a typical turtle) would need about 75-125 gallons, though more would be better. You could probably get by with a 40 gallon breeder tank for a few years when it's young, but would need to eventually upgrade.
There are smaller turtles, such as musk turtles, which could actually do well in a 20-30 gallon long tank (they like it shallow). They also have slightly easier maintenance (don't need as strong a basking spot). They don't usually have as distinctive of patterns as other turtles, but they still can be pretty neat -- their long necks and small plastrons make them nimble little climbers. I'd probably recommend one as a starter turtle.
There's also non-fishtank options, such as turtle tubs (either pre-made, or built yourself) that are just wide tubs with sides just tall enough to keep them from climbing out. These can sometimes be cheaper to acquire than 90 gallon aquariums, but take a lot more floor space.
Whatever way you go, you'll need to invest in a good quality, big filter as well. They're very messy, and constantly cleaning an underpowered filter will get old really quickly.
This page seems to have a lot of good stuff too.