Each player starts the game with a deck of ten artichoke cards, with five of those cards in hand. Five garden cards are displayed face up in the garden row next to the garden deck. On a turn, you draft a garden card into your hand, play all the cards in your hand that you wish (placing them in the discard pile as soon as you finish an action), discard any cards still in hand (such as artichokes, which do nothing on their own), then draw a new hand of five cards. If at this moment you have no artichokes in your hand, you win!
Like many deck-building games, Abandon All Artichokes starts simple, then you scale up in difficulty — maybe. On your first two turns, you typically have only one action each turn, that is, whichever garden card you just picked up, and depending on which card that is, you might compost that card along with an artichoke or two, leaving your deck somewhat lighter in artichokes, yet not changing the artichoke percentage of your deck.
In general, you have two courses of action that will lead to victory: (1) Strip your deck of artichokes so that you can't draw any, and (2) Add lots of garden cards to your deck (while perhaps also ridding yourself of artichokes here and there) so that your in-deck artichoke percentage falls, thereby giving you a chance of drawing five garden cards.
The garden cards are split in terms of whether they help one course of action or the other, but they also vary in their effectiveness during the early, mid-, and late parts of the game (as quick as those are in a 10-20 minute game). Pepper, for example, places a card from your discard pile on top of your deck, which means it does absolutely nothing on your first turn and little on your second, yet later in the game you can set yourself up to ensure success from a Potato (which eliminates the top card of your deck if it's an artichoke) and should you manage to have lots of Pepper in hand, you can stack your deck with multiple garden cards, which means you're relying less on luck to win since you now need fewer non-artichoke cards to draw your way to victory.
I've played Abandon All Artichokes ten times on a review copy from Gamewright, mostly as a two-player game with my exchange student Lisa, but also with three and four players. In addition to certain cards being better early than late and vice versa, the cards also change value based on the player count.
Peas, for example, lets you draw two cards from the garden deck, then place one in your discard pile and the other in somebody else's. In a two-player game, you're getting the card of your choice while the other player gets randomness, yet you're both reducing your in-deck artichoke percentage, so it's kind of a wash; with more players, you can spread the benefit to multiple players, similar to how in Bohnanza you want to trade with as many people as possible to maximize your benefit while spreading around the pluses to others.
That said, Lisa loves playing Peas and she beats me consistently in two-player games, so my disdain for Peas might be soured by spite. For more on the game, watch this video overview: