I’ll be the first to say it: you really don’t need all the knives in a set. Good cutlery is a lot like good cookware - buy only what you need, when you need it, but avoid the sets as they include things that you’ll never use.
I recommend getting yourself a good quality chef’s knife, probably 8 inches long. (I think the 6" are too short; the 10" too long.) Go to a dedicated cutlery or cooking store and try it out - you want to get something that fits your hand and feels good. The fit and feel is more important than the brand, really, and don’t be afraid to mix and match. The more expensive brands will cost you $80 - $100 for a good chef’s knife, tho’ I’ve heard of cheaper ones that have gotten a number of recommendations from serious cooks, like Cooks Illustrated and such - maybe someone will pop in with a link. Myself, I prefer Wustof. I have some of their classic line. I use my chef’s knife for easily 90% of all my prep in the kitchen - it’s really worth the money to get a quality one.
A knife that’s often recommended as the next purchase is a paring knife, but I kind of disagree. I have one, but I rarely use it. (Probably because I don’t peel much of anything - the potatoes are unpeeled, the cukes unpeeled, and so on. Can’t remember the last time I peeled anything. And then when I do, I usually use a veggie peeler - never got the hang of peeling with a knife, really.)
After that, it varies. I’d get a nice breadknife and maybe a cleaver (ooh - or maybe a tomato knife. That’s probably the knife that does the other 10% of the cutting in my kitchen), but others might recommend a boning knife or steak knives or what have you. That said, if you can find a set or block that includes only 3 knives (like the chef’s, paring, and bread or some such), it might be worth it. Just don’t get a set that includes 10+ knives, as there’ll be a bunch in there that you probably won’t use. Start small.
One thing that is worth the purchase: kitchen shears. I bought the Wustof shears and use them constantly. And they weren’t expensive; maybe $10? I know they’re not of the same quality as my knives, but they’re shears - they’re meant to get beat up and be replaceable.
I’ve heard that knives should be resharpened every year. You can buy sharpeners for your knives that take any guesswork out of it and help maintain the edge, but it’s probably worth getting them professionally sharpened every 2-3 years or so. (That said, I haven’t gotten mine resharpened, and it’s going on probably 5 years now. I need to do that, obviously.) Using the steel will help maintain the edge as well, but I have to admit that I’m not so great at that either.
Other than a tomato or bread knife, stay away from serrated edges. Serrated knives are difficult, if not impossible, for home chefs to resharpen - you end up just grinding down the serrations. They’re great for tomatos and bread, but regular old knives will work for everything else. And your tomato and bread knife are probably best served by being professionally resharpened.