I spent years avoiding knitted tops because the ones I owned pilled under my arms within a month and clung to every bit of me I didn't want clung to. What changed was finally trying one in a slightly heavier gauge instead of the thin, see through stuff, and realising the problem was never the idea, it was the version I kept buying.
The honest limits first. A fine, floaty knit shows every seam of whatever bra you're wearing underneath, so if you're not going to wear a smooth one, pick a thicker knit or accept that everyone can see the lace trim. Knitted tops also aren't brilliant for anything with a bag digging into one shoulder all day, the fabric drags and goes baggy in that one spot and never quite recovers. And if you run warm, a lot of these are not breathable in the way a cotton tee is, so for actual hot weather or anywhere stuffy, I'd leave it at home. For fit, go up a size rather than down if you're between two, because knit clings and a size too small will show at the stomach in a way a woven top never would, while one true to size or slightly bigger skims instead. Watch the shoulder seams, if they sit halfway down your upper arm the whole top will look like it's slipping off, and check the sleeve length sitting down as well as standing, since a lot of these ride up at the wrist the second you lift your arms. The hem is the other thing to feel for in the changing room, a knitted top with no weight to the hem will ride up over trousers all day and you'll be tugging it down on repeat.
On colour, I've had the most use out of oatmeal, a proper navy and a soft burgundy, none of which show deodorant marks or pet hair the way black does, and all of which go under a blazer or over a shirt collar without looking like you're trying too hard. A plain cream one with jeans and boots has become something I reach for without thinking, and the same top tucked into a skirt does a completely different job for very little effort. Cable knits in particular hide a multitude, which is probably why half my friends own the same three colours and swear they didn't plan it that way.
Next time you're looking, feel the hem first, if it has a bit of weight and drop to it, the rest of the top is usually worth trying on.