What can be grown depends a lot on where you are on the island as well as which variety of what you are growing. We have loads of food from the yard but we have a fairly large yard by Oahu standards although it is a small yard by Hawaii Island standards.
A lot of folks put in trees and semi-permanent plants so they can get food without a lot of effort. Citrus are good trees for smaller lots, especially the dwarf varieties. They will grow up to about 1,800 foot elevation, I think. If you are planting a citrus, look that number up, I'm just guessing. Elevation makes a huge difference on what you can grow, although Oahu doesn't have the same elevation differences of some of the other islands.
For a smaller yard at less than 1,500 to 1,800 foot elevation, these are some of the things which might do well:
1. any dwarf citrus - oranges, tangerines, lemons, etc. Water the new trees fairly frequently (deep watering about twice a week) for the first year or so and add a lot of compost and bunny manure to the hole dug for the tree if you can. Adding a top dressing of composted bunny manure is good, too. It might be worthwhile to get a pet rabbit just to get bunny manure.
2. Bananas. There are loads of different varieties to choose from, we plant the dwarf Chinese variety because it produces a lot on short plants. When the stalk produces fruit, chop it down - it will die off anyway so you may as well) and then leave the chopped trunks around the base of the other bananas to mulch them. Or put the trunk in the compost pile or use it in an imu. It is extremely hard to overwater or overfeed bananas. Water them, give them mulch, compost, bunny manure, etc. They also prefer to be out of the wind if they can although they will tolerate if they have to. It shreds their leaves, though.
3. Papaya. They are easy to start from seed, buy a papaya to eat and plant the seeds. You can also get seeds from CTAHR
Agricultural Diagnostic Service Center - Seed Program for about $1 per packet. Papaya are pretty easy about where they will grow although they do best with mulch and water. Don't pile the mulch around the stem of the plant, though, or that could cause rot. There are male and female plants so having several of them around are good. Usually someone close enough in the neighborhood will have a male plant, though, so just one female one will be enough if you don't have a lot of room. Papayas and bananas are a temporary plant, they will last for a few years and then die off. Bananas usually have a lot of keiki which come up from the roots, though, so you'll end up with a banana patch where the first banana was planted. Papaya, you need to replant about every six to ten years. Citrus are good for decades.
4. Passion fruit vines. They are incredibly vigorous, so have plenty of room for them to grow. Buy a passion fruit and plant the seeds.
Herbs are pretty easy to grow, either in the ground or in pots. They usually like good garden soil and good drainage. We like to put them within easy access of the kitchen. For a sunny somewhat drier spot, you can choose plants such as chives and rosemary. For a sunny spot that is wetter, green bunching onions, basil, oregano, marjarom, etc. For a damp spot in semi-shade, the mints will be happy.
We stacked concrete blocks two rows high in a shape about four feet wide by eight feet long and filled that with whatever topsoil we could find. We added a lot of semi-composted shredded tree bark from a pile that had been there for about six months and added about four five gallon buckets of bunny manure. Then we planted assorted vegetables such as lettuce, onions, beans, tomatoes, watermelons, etc. The results have been shocking and I'm contributing that to the bunny manure. I've gardened for years and never had such spectacular results.
Hmm, I think I'll go offline and go fuss about the garden now, come to think of it. One of the watermelons is probably ripe. Oh, we also set out fruit fly traps to keep them from "stinging" the fruit. We did have to hand pollinate some of the watermelons since there didn't seem to be many bees this year but there seem to be a few now.
Check out the Hawaii section of GardenWeb for tips on gardening in Hawaii or check out the CTAHR site, although that can be hard to navigate. Tons of information if you can find it, though.