Trick-taking games
Nearly every serious card game in the Western world is a trick-taking game. One card each, highest wins, next player leads — and everything interesting happens in the constraints layered on top.
The core is almost trivially simple. Everyone plays one card. If you hold the suit that was led, you must play it. The highest card of that suit wins the trick, and whoever won it leads the next one. Learn that and you can sit down at any game on this page.
The first great complication is trump: a suit that beats all others regardless of rank. It arrived in fifteenth-century Europe with tarot decks and it transformed the family, because it means a two can beat an ace and being void in a suit becomes an asset rather than a weakness. Spades fixes trump permanently. Euchre and Pinochle let you choose it. Bridge auctions it. Hearts, pointedly, does without it altogether.
The second complication is what you are actually trying to achieve. Most of these games ask you to win tricks — but Hearts inverts the whole thing and makes every trick a potential liability, which turns out to be just as deep and much stranger to learn. Spades and Bridge add a promise: you declare in advance how many tricks you will take, and winning more than you said is a problem too. Pinochle bolts on a second scoring system entirely, awarding points for combinations in your hand before a card is played.
Where to start depends on what you want. Hearts if you want to be playing in five minutes. Spades if you like arithmetic and have a partner. Euchre if you want a hand to last two minutes. Bridge if you want a game you can spend years on and still be improving. Pinochle if you want something most people you know will not have heard of.