Count me as another in the three-knife category (I prefer a traditional Euro/French style knives)...
8"-10" Chef's knife - Most of your major slicing and veg. chopping. It's pretty much the workhorse of my kitchen.
6"-8" Boning knife - Deboning chicken, fine slicing of meats, and can even be used as a carver or filet if need be.
3"-4" Pairing knife - Does all your finer detail work like breaking down small items, pealing, etc..
If you're buying name brand, you can get all three for well under you're $300 budget. I bought my set of Henkle Pro's as a 3-pack at one of the big kitchen stores for around $150-ish about 15 years ago, and will never go back to a cheaper (quality) knife. But you can also get Wusthoff, Shun, or Global well within your budget for all three knives. Just try them in hand first to see what you like (Global's handles don't work for me). Most of the higher-end kitchen stores have demo knives you can try.
Vitrinox also makes a good knife for the money, and you'll be out the door for a third of your budget.
Also remember to budget in a good steel and a real wood cutting board, if you don't already own them. You'll still be well within budget.
Sorry, can't help with Japanese style knives, since they're not my preferred choice. Also don't have any experience with custom.