The best knife to use is the one that you can wield comfortably, effectively, and safely. Personally, I'd use a 8" chef's knife for most of the items on your list, and I'd guess that a similar knife would be a workhorse in many kitchens, but that doesn't mean that that knife is the right one for you.
Good books on cooking (not necessarily a "cookbook") will often have information on developing your knife skills. The New Professional Chef is one book that has such information. There are also a number of videos on YouTube that demonstrate good technique. It's fine to read about or watch knives in action, but the thing that really makes a difference is practice. Buy a few pounds of celery, a big bag of onions (and Kleenex!), a pile of carrots, mushrooms, and potatoes. Practice the various cuts, slowly at first, until you feel comfortable and competent. Most of all, practice keeping the fingers of your holding hand tucked underneath and out of the way of the blade.
To answer your specific question, a decent straight-bladed knife somewhere between 7 and 10" should be fine for slicing, dicing, or chopping peppers, onion, chicken (the raw and the cooked), tomatoes, carrots, and the rest. If you want to do something like taking the core out of a tomato without first cutting it in half, use something smaller like a paring knife. You'd also use a paring knife when you want to make smaller, more controlled cuts, like peeling a fruit or vegetable. A large serrated knife is great for slicing bread, and a smaller serrated knife can be handy for fruits and vegetables that might have a slippery skin, tomatoes particularly. (If you keep your knives sharp, and you should, a straight blade is fine for tomatoes; if not, the serrated blade helps a lot.)
Those are the basics. The name of a knife usually gives a pretty good indication of what it's for: a boning knife for boning, a carving knife for carving, etc. Don't feel like you need to collect the whole set right away.
One more thing: Get yourself a good pair of kitchen scissors, i.e. one that easily disassembles for easy washing. I use my scissors for everything from snipping fresh herbs to wreaking havoc on whole chickens.
Still one more thing: you also need decent cutting boards. Decent doesn't mean expensive, just flat, large enough (bigger than the food, bigger than the knife you're using), and unlikely to rock or slide around on your countertop. If sliding is a possibility, put the board on a dish towel to stabilize it. Go for wood or plastic to protect your knives, not something hard like glass.