Aquatic turtles need a complete diet with trace nutrients and vitamins. The simplest way is to try some of the commercial turtle pellets available such as reptomin, zoomed, and mazuri. You can then supplement this with worms, insects, and dark leafy vegetables (for most pond/river species).
Tortoises are similar, but their diet should be composed of approximately 90% dark leafy greens (kale, mustard, dandelion, etc.) and 5-10% fruits and fleshy vegetables. A good supplement is a commercially prepared tortoise diet like Mazuri tortoise chow. For larger tortoises, they may eat it without water, but smaller tortoises try soaking the pellets in warm water for 60 seconds, drain, and offer to your pet. These pellets are a great way to get extra vitamins and calcium necessary for proper growth and health!
Terrestrial turtles will benefit from extra protein like worms, slugs, snails, and cooked chicken or fish.
Also, it is important that all turtles and tortoises have access to UVB lighting. We recommend a google search on "UVB lighting for reptiles" to find a bulb that fits your needs/budget. Or better yet, keep them outdoors as weather permits in your area!
Please remember these are GENERAL recommendations for most common species of pet turtles and tortoises. For all aspiring turtle keepers, it is important to research your specific species online and in books. Your local library can be a great resource for reptile husbandry!
The California Turtle and Tortoise Club is a good start!