It'll come down to personal preference I should think, but I find just using a good quality, 5 inch utility knife works well for most day to day stuff like filleted meat and veg (though nothing like a melon!). It's a nice size to handle easily, without being as small and fiddly as a paring knife. Having a full tang (where the steel that forms the blade runs the full length of the handle as well) feels sturdy and robust too.
I'll add in a proper chef's knife at some point too, though I'd find it a bit oversized to use for everything.
Serrated can be good for things like tomatoes, although a properly sharpened straight blade will work just as well. Similarly, Germans think we're mad for using serrated bread knives since they tug at the bread, and a sharpened straight blade will glide right through. Keeping them sharp will also cause less strain on your hands from trying to wrestle through things.
Very much not an expert, but some good brands are Wusthof, Zwilling J.A. Henckels, Global and Robert Welch. Not cheap, but expected to last for donkeys. Harts of Stur are usually good on prices if you want to have a peruse of different brands, and they run the gamut from "cheap as chips" to "sell a kidney".