Tomato- Cherry, Grape, Celebrity, Early Girl, Better Boy
Snap Beans- Bush type
Peppers-Jalapeno, Serrano, Habanero, Anaheim, Poblano, Thai Hot, Hungarian Wax, California Wonder, Most bell peppers.
Generally speaking you will need a 5 gallon pot for each plant. Root vegetables will need to be a longer 5 gallon window box being at least 12 inches deep. Plastic, ceramic, terra cotta, wood are all fine choices for containers, however they must have drainage holes in the bottom. You'll also want to set your containers on blocks or bricks so the water will drain freely. If you've never grown anything, I'd suggest getting the Bush type Snap Beans, and a Tomato to try out. Lettuce is fairly easy but in both Ohio and California (the 2 places I've gardened) you're going to have problems with slugs and snails who will devour leafy veggies overnight.
When potting your vegetables I suggest getting an organic potting mix. If you're not trying to be organic that's ok, Miracle Gro potting mix will work fine also. I'll type this next line in all caps....because it's important....haha. DO NOT USE MIRACLE GRO GARDEN SOIL IN A CONTAINER OF ANY KIND. I say this because Miracle Gro Garden Soil confuses everyone. It says "Garden Soil" so it must be good for the garden right? It is good for the garden, if you're planting directly into the ground. This particular soil is mixed into the soil in the ground. It contains manure, if you try to use it in pots, the manure will burn up your plants and they will all be dead in a week. As long as it says "potting mix" or "potting compost" or "potting soil" it will be fine to use in your containers.
Watering is the next step. Plants in containers will dry out much more rapidly than the ground. Always water in the morning. Plants need the water throughout the day for photosynthesis. It isn't a good idea to water in the evening because the cooler temperatures cause the water to evaporate at a slower rate. This can cause disease, and root rot. Both will kill your plants. You'll want to water the plants thoroughly, not sprinkle them. Sprinkling a small amount of water on the plants can cause the roots to surface, and the sun exposure will kill them. In the heat of the summer in California, you'll want to water your plants every morning. Tomatoes and peppers can handle the heat in California. If you've got herbs, place them where they'll receive morning sun. If the temperatures rise to 100 degrees your herbs will burn up in a matter of hours. Bring them inside if you've got nowhere else to put them in the afternoon. In Ohio you might not have to water every single day. If you're uncertain about when to water you can buy a soil moisture meter. They range in price from about $7 on up into the hundreds. Here's one made by Rapitest that works well, and requires no batteries. I've used this one and so have several of my gardening buddies. Its also available at most stores in the garden section.
Fertilizing is next. I recommend getting an organic liquid fertilizer that can be diluted in a watering can. There are hundreds of kinds just look in your nursery for one. Plants in containers need a more regular fertilizing routine. If the potting mix you bought isn't enhanced by Miracle Gro or some other fertilizer you'll need to fertilize once a week when you water. If the soil you buy already has fertilizer in it, I wouldn't bother fertilizing for at least a month. This goes for all vegetables except hot chili peppers. If you are growing chili peppers for the heat, I suggest planting them in black plastic containers. The black containers hold more heat, and there is a direct correlation to the soil heat, atmosphere heat and chili heat. Also, once the pepper produces its first flowers stop fertilizing it. This will make the peppers hotter. Allowing the peppers to dry (to a slight wilt, SLIGHT) between waterings will also create hotter peppers. I know this sounds scientific but it's something I learned last year. None of my chili's were hot. This year...they will be.