Turtles need a few things to thrive- space, warmth, sunlight, clean water, and good quality, varied food.
Space: Aim for at least 10 gallons of swimming space per inch of water.
Warmth: Heat the water to 75-80F, and the basking site to about 90F.
Sunlight: Turtles NEED UV-B rays for health, and lighting that simulates natural sun. A good reptile bulb that advertises UV-B is a simple choice here.
Cleanliness: Aim for a filter that is about 2-3 times more powerful than a fish tank would use.
Food variety: Try a mix of about 25-50% of a good quality pellet and the rest live or frozen/thawed 'fish food' like small fish, worms, bugs, shrimp, beef heart, etc. Small turtles usually like blood worms.
They often go through a short (2-3 week) adjustment period that is no big deal if the temps, etc. are good.