A good coaster sits flat on the table without rocking, grips the bottom of a glass so it doesn't skate across the surface when someone bumps the table, and takes a sweating cold drink or a hot mug without flinching. You should be able to pick it up after and not find a ring left behind on the wood underneath. That's the whole job, and it's surprising how many of them fail at it within the first month.
Cork and stone ones can scratch a softer wood tabletop if you drag them instead of lifting them, so on anything with a delicate finish, pine, reclaimed wood, anything oiled rather than lacquered, go for a felt backed coaster rather than raw stone or unfinished cork. Metal coasters look sharp on a glass or marble table but they're a bad match for wood, since any grit caught underneath turns them into sandpaper the moment someone slides their glass. And patterned ceramic ones that look lovely in a kitchen can disappear or clash on a table with a busy wood grain or a heavily patterned runner underneath, so keep them plain if the surface they're sitting on already has a lot going on.
Size matters more than people think. Measure the base of your largest mug and your widest wine glass before you buy a set, because a coaster that's slightly too small will let the glass overhang the edge, and that overhang is exactly where condensation runs down and pools on the table anyway, defeating the point entirely. Four inches across covers most mugs and tumblers, but if anyone in the house drinks from oversized bowls or stemless wine glasses, go closer to four and a half. The other mistake is buying a set that's too thick and stacks awkwardly in a drawer, or too thin and soft so it slides around on a glossy table instead of staying put.
They're not going to protect a table from a hot pan or a dish straight out of the oven, that's a trivet's job, and asking a coaster to do it will leave you with a warped bit of cork and a table with a mark anyway. They also won't do much for a table that already has a cheap finish wearing thin, since the real risk there is the daily use, not just drinks.
A set that matches how you actually drink, in terms of size and surface, beats one that just photographs well on a shelf.