Another good climbing game is Clubs. It's super-simple so you can teach it to anyone, but it has a surprising amount of depth and strategy. It's a commercial game with a nonstandard deck, but you can also adapt it to use a standard deck without losing anything important from the gameplay.
BGG page:
https://boardgamegeek.com/boardgame/125924/clubs
For a standard deck, instead of fifteen cards 1-15 in four suits, you'll have 13 cards A-K (ace low) in four suits. It'll still work fine.
Rules in brief:
3-5 players. Deal each player ten cards and set the rest aside. You can also play with six players and deal each player eight cards. The player on the dealer's left will lead to the first trick.
The two goals are to a) go out (play all of your cards) before other players and b) capture Clubs cards in tricks you win.
When you lead to a trick, you can lead a set of cards (a meld) consisting of either a) one or more cards of the same rank or b) two or more cards in a run (suits don't matter). In turn after that, each player on their turn can either play or pass. To play you must play a meld of the same type and size as the lead; i.e. play pairs on pairs, play trips on trips, play a run of two on a run of two, etc. You have to play a higher
rank; i.e. a pair of sevens on a pair of fours, or a run of 6-7-8 on a run of 4-5-6. You can't change the size of the meld, i.e. you can't play quads on a pair, and you can't play a run of four on a run of three.
The trick continues around the table with players playing or passing until
all players but the current trick-winner pass. This means that the trick might go around the table several times, unlike in standard trick-taking games where it only goes around once. Once everyone else has passed, the trick winner (i.e. whoever made the last play) takes the trick and leads to the next one. If any clubs were played on the trick, the winner keeps them, and will score points for them at the end of the hand.
If you don't have a valid play, you must pass. You never have to play; you can always pass even if you have a valid play. If you pass during a trick, you can still play on the same trick later if it comes back around to you.
Anyone who plays their last card is out of the hand; the rest of the players continue playing without them. Players who go out get bonus points; the sooner they go out, the more bonus points they get.
The hand ends when only one player is left with cards. That player scores zero points; they get no points for clubs even if they've captured some, and they get no bonus points for going out. The other players score points for the clubs they've captured and bonus points for going out early.
Scoring:
Clubs captured:
- Ace: 5 points
- 2/3: 4 points each
- 4/5/6: 3 points each
- 7/8/9/10: 2 points each
- J/Q/K: 1 point each
Going out early:
- Last out (last player still with cards): 0 points, and no points for clubs
- Second-last out: 2 bonus points
- Third-last out: 5 bonus points
- Fourth-last out: 8 bonus points
- Fifth-last out: 10 bonus points
- Sixth-last out: 12 bonus points
So, for example: the first to go out in a four-player game will get 8 bonus points.
Double-or-nothing: A player with a strong hand can declare "Double or Nothing" anytime before they play their first card in the round. They are thereby betting that they will be
the first player to go out that round. If they succeed, they score double points both for their clubs and their going-out bonus. If they fail, their score for the round is zero, no matter when they go out or how many clubs they take.
Give it a try! It's fast, easy, and fun, but requires some tactical thinking and awareness. There's a fair degree of luck in any single hand - strong hands will score high, weak hands will score low - but if you play several hands the luck of the draw will average out and the better players will come out ahead. For games like this I like to play one hand per person, i.e. one "orbit" so that everyone has a chance to be the first on lead. At the end of one orbit tally up the scores, and if you're feeling gambly then have the losers pay the winners based on the difference in their scores.