What are the purposes of the various specialty knives?

The standard answer is that you actually need a grand total of three knives, and could conceivably make do with one. The usual load-out recommended is 1)6"-8" chef's knife, santoku, or cleaver, 2) 3" paring knife, and 3) long serrated knife. The chef's knife or santoku is for most work (a cleaver can be used for this if you're a mad man), the paring knife is for small work (fine vegetables and mushrooms), and the serrated blade is for crusty breads and tomatoes (which can end up crushed by straight blades). If you are embarrassed for funds, ditch the serrated knife and paring knife, in that order. If you're doing serious butchery, a 10"-12" carving knife and a small thin jointing knife are worth the investment. There's a special knife for fileting fish, but to be frank I've had one for 20 years and it's never been the best tool for the job.

Watermelon can be sliced with a large straight blade (I might pull out the 12" for that), but only with a lot of gumption. Show fear or hesitation, and you're going to make a mess.

More important than what you buy is how you use it. Search on youtube for "introductory knife skills", or take a cooking class on it. Well worth the investment.

(As should be obvious from the above, whatever you do don't buy a "knife set". You'll end up paying a lot of money for knives you will probably never need.)

Except for raw tomatoes, french bread, and peeling fruit, everything on your list ends done by my 8" santoku (or 6" ceramic, which is not something for beginners, purely on price).

Okay, I'm going to partially answer the initial question / title, as the pairing really isn't what knife for what food, but what knife for what type of cutting. And I say partially answer, as there are a lot of types of knives ... I'm only going to list the more common knives, as the japanese have a whole lot of specialty knives.

This is only the tip of the iceburg. I'm making this answer community wiki, if people want to add knives, or details of anything I listed. (although, if you want to organize it by use (eg, chopping/slicing/paring/etc, it might be better to move that to a separate answer, or this is going to get way too long)

• Chef's knife -- has a large-ish blade (generally 8" or longer, ~2" high near the bolster), with a convex curve that's more dramatic near the tip. Its characteristic use is to keep the tip on the cutting board, and rock the handle down (preferably using the weight of the blade) to slice through things.

• Paring knife -- a smaller knife (~3-4" long), used for paring. (cutting in the hand; peeling, stuff that doesn't require a cutting board). The blade may be concave (bird's beak), straight (sheep's foot) or convex (spear point).

• Slicer -- a long (~10" or longer), straight, thin knife, used for slicing large items (eg, meats)

• Bread knife -- a long (~8"+) serrated knife, used for slicing bread.

• Boning / Fillet knife -- a thin, flexible blade, used for cutting between joints and stripping meat from carcasses.

• Meat cleaver -- a thick, wide (often 4"+ tall) knife, used for hacking through bones or frozen items.

• Vegetable cleaver / Chinese cleaver : a sharp, thin, smaller cleaver, used for delicate work that should never be used for cutting through bones.

• Santoku -- cleaver-like in a design, but has a slight convex blade and closer in scale to a small chef's knife (5 to 7" long, about 1.5" tall). Good for general use.

• Utility knife / tomato knife -- mid-sized (~5-6" long), serated knife. Good for when you need a small serated knife, such as um ... tomatoes. And works pretty well for cutting medium firm cheeses.

• Cheese knives -- um ... see this answer on cheese cutting

• Steak knives -- mid-sized knives to be used at the dinner table, that are either very, very sharp, or serated.

• Table / dinner knife -- the basic sort of knife as part of a table setting

• Butter knife -- a short, dull knife, possibly offset, used for spreading butter or things of a similar consistency.

And not knives, but related:

• Kitchen shears -- look like your general run-of-the-mill scissors, although the good ones come apart so they can be cleaned easily.

• Poultry shears -- heavier duty, often spring loaded ... look more like pruning sheers (that you'd just for trimming down a rose bush). They have a notch in one of the blades near the pivot point so that you can cut through a bone without it slipping. Ideal for cutting the backbone out of a chicken.

• Bench knife / scraper -- a wide, dull piece of metal with a handle along one edge. It's used for dividing dough, scraping down your work surface (bench), but can also be used to scoop up stuff to transfer to another location (eg, dump in a pot)

• Pizza cutter -- a circular blade attached to some sort of a handle (which might be more like a shroud). Can be used for lots more than just pizza, though. Great for cutting thin, but large objects. (eg, cutting breadsticks, trimming filo, puff pastry or other rolled out dough, etc.)

• Vegetable peeler -- a handle with a swiveling sort of blade on it, but where the cutting edge is along a slit in the middle. You have standard peelers, and 'Y' peelers. Normally smooth blades, but may be serrated. Good for peeling the skin off items, cutting vegetables into ribbons, or shaving chocolate.

• Mandoline -- a blade fixed on a planar surface. Used to make thin slices of a consistent thickness by moving the food against the blade.

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.

I found this page useful.

Myself, I use a 20cm chef's knife for most tasks. It is designed to be general purpose. Just remember it needs to be sharpened from time-to-time. And it needs to be sharp.

For some things (e.g. cutting tomatoes), a small serrated knife is good, but a really sharp straight knife will also work.

The only other knife I use regularly is a bread-knife.