In general a turtle's diet is comprised of 50% protein, 30% vegetables, 10% greens, and 10% fruit. The protein can even be divided up so that it's 25% commercial food, and 25% fresh food.
Either Zoo Med Aquatic Turtle Food or ReptoMin Plus are the most popular brands of commercial turtle food that you can find in a pets store where I live. Personally I prefer Zoo Med as their focus is on reptiles.
Since your turtle is really young, you'll also want to feed it calcium supplements - Zoo Med also makes a "Turtle Bone"". That will help the turtle's shell and beak grow healthily (avoiding bone/shell deformities).
You can feed it live food such as:
- Worms (e.g. earthworms or waxworms) But only if your pet store sells them. You won't want to pick earthworms from outside, as they could have been exposed to pesticides, and bait stores don't sell them as food so who knows where they've been.
- Fish like feeder goldfish or guppies. Guppies are better in my opinion, as the feeder goldfish have spiny fins.
Boiled chicken is also a really good staple for protein.
Vegetables/greens include:
- Peas
- Green Beans
- Summer Squash
- Potatoes
- Okra
- Carrots
- Collard, Mustard, or Dandelion Greens. (No dandelions from outside because of possible exposure to pesticides and parasites)
You can also feed it fruits on occasion. I say they're more of treats because of their generally high water content vs actual nutrients though. They do make a good way to encourage your turtle to eat if it gets finicky. Just mix in some of it's favorite fruit in with some other food. You can also use fruits as a way to administer vitamins too.
I don't think there are any fruits that are really bad for turtles (aside from low nutritional value). The most common I know being fed to turtles are: Grapes, blackberries, cherries, cantaloupe, bananas, apples, kiwis, etc.
You can also feed it treats like dried prawns.I consider these treats because they're so high in protein, and so low in other nutrients, that they're not really good as a staple food in my opinion. The goal is to have to give as little vitamin supplements as possible. There is also the risk of your turtle liking the taste too much, and refusing other foods that it needs, like the vegetables. Kind of like a kid given a choice between green beans and candied bacon, it's always going to choose the candied bacon. But you can mix it in with other foods as a supplement if you want to, and definitely if it's something that your turtle might not want to eat by itself.
You can also feed it live prawns, which give the benefit of the mental stimulation from hunting.
Other treats you can give it include mushrooms, corn on the cob, low-fat dry dog food, which is a good source of protein, but not really anything else for a turtle, and on a rare occasion a boiled egg or pinky mouse, though not at the size of your turtle right now.
Foods you never want to feed it:
- Lettuce/spinach
- Avacado
- Processed meats
- Meats high in fat or salt
- Raw meat
- Rhubarb
- Tobacco leaves
- Things that only humans should eat (Sandwiches, hot dogs, chili, mac & cheese, etc.)
Note that they can be messy eaters, so you'll want to make sure that you clean the tank after feeding times if you don't feed it in a separate tank.
As a side note, since I'm assuming this is a wild turtle you caught. I'd suggest looking into your local/state laws on keeping native species. I know in most midwestern states, it's illegal to keep most native turtles as pets. In some it's actually illegal to even pick up the endangered species.