Stand in front of a vanity or a basin and put your hand flat on the edge of it before you decide anything. That's the part nobody tells you to check, and it's the part you'll touch every single day for the next ten years, so if it feels thin or hollow or sharp at the corner, walk away no matter how nice the color is.
If you're starting from nothing, I'd get the vanity sorted first, since it sets the height and the storage and basically dictates what basin and faucet you can even use with it. A simple vanity with real drawers, not a cabinet with one shelf behind a door, will do more for a cramped bathroom than any amount of clever decor. The sink and faucet can follow once you know the footprint, and the toilet is the one piece I'd never get precious about, since a plain, well proportioned one in white will always look right and is the easiest thing in the whole room to replace later if you change your mind.
What separates a good one from a sorry one usually comes down to what you can't see at first glance. A solid, properly glazed basin should feel cool and dense under your hand, with no rough patches where the glaze skipped, and the overflow hole finished as cleanly as the rest of it. Vanities are the same story, check the drawer runners slide without rattling, see if the cabinet feels planted when you lean on it rather than rocking on one foot, and look at the back panel, cheap ones are open board that swells the first time water gets near it. Faucets live or die on the handle action, a good one turns with a quiet, even resistance, a bad one feels loose or sticks halfway and will likely drip within a year.
For color and finish, I'd stay with white or a soft off white for the basin and toilet almost every time, since they're the hardest things to replace and the easiest to live with against any wall color you choose later. Where I'd spend a little nerve is the vanity and the faucet, a warm wood tone or a muted sage or clay colored cabinet against a plain white basin looks settled rather than show offy, and brushed brass or matte black hardware reads warmer than shiny chrome in most ordinary bathrooms. All of it sits easiest with plain tile, a simple mirror, and one good towel rather than a lot of pattern competing for attention.
If I had to pick one room to get right first, it's the main family bathroom, since that's the one every person in the house and every guest will judge the whole place by.