I'm new to gardening, what veggies grow best in large gardening containers?

bell peppers!

Tomatoes do quite nicely. You can add in some basil, oregano, parsley, and/or thyme and have a nice little pizza planter. The basil will really help the tomatoes do well.

Thank you

Things like carrots or potatoes do well in deep containers, like feed buckets. I have lettuce planted in an old wheelbarrow in the yard. Check out gardeners.com for some innovative products for container gardening. They have these things called grow bags that look really interesting.

I second tomatoes and bell peppers. Also herbs, which can really improve the taste of food.

I have grown things like, tomatoes, peppers, eggplant, zucchini, squash, cukes, peas, and lettuce. This year I am trying carrots, swiss chard, and spinach which are all doing quite nicely in the containers.

Tomatoes, and add to what @MyNewtBoobs said some garlic in the pot.