Hi CS, just a home cook here, but I concur, very good and cheap do not usually go hand in hand.
Sometimes it is cheaper in the long run to get something a bit more pricey than to keep buying replacements.
But you can get some decent knives fairly inexpensively. We have sort of a collection we put together over the years. Never purchased a very good/expensive one. (Wait a minute, I have a birthday coming up pretty soon, hmmm, I have always wanted a good one).
I find the one(s) I use the most, OK, read that almost always, is a chef knife.
The one I use the most has a blade of about 6".
As important as the knife, you have to know how to keep it sharp.
You can Google for instructions. A sharp inexpensive knife works a heck of a lot better than a very expensive, but very dull one.
(Generally the more expensive ones hold their edges longer).
If I could have only one knife it would be a chef knife.
Then a paring knife. And then, you mention peeling, I just use a relatively inexpensive peeler (know that may not be the professional way to do it, but I find it easiest).
Glad you like cooking and have a very happy birthday.