Right up front, I should say I own more copies of chess and have played it more than anything else in my library. I enjoy teaching chess whenever the opportunity presents itself and I occasionally get into a chess kick where I play against one of the many available AI opponents. It's a game that I sometimes tire of, but always return to.
I won't go into detail about the game's mechanics since there are many places to find them. (I recommend Bruce Pandolfini's Beginning Chess or Bobby Fischer Teaches Chess which both include enjoyable chess puzzles as well.) There's no point in reviewing game components and such either since there are so many varieties for sale. (My only recommendation is to start with weighted plastic pieces of the traditional shapes so that you can follow chess diagrams.) This review is more about how to play a game with a centuries-old metagame.
Chess is a completely open game, which means all possible moves and counter moves are known right from the start. White begins with 20 possible first moves and only about half of them are viable. Black must respond with the same 20 moves. Within a few moves, the possible board positions are astronomically huge. Most of those positions can be shown to be advantageous for one side or another and so there are a limited number of viable openings. Unfortunately, a fraction of a huge number is still quite large and it's not always (or even usually) possible for a beginner to find the optimal move.
And that's just opening theory. There's tons to learn about the endgame (when each side is down to a handful of pieces) and midgame tactics (forks, pins, skewers and so on). Player who have studied chess theory have a significant advantage over those who have not. There are few things more dispiriting than being outmatched in a game of chess. This probably explains why there are so many chess sets that haven't been played and so many players who give up after one or two matches.
Generally modern game design strives to protect less experienced players from discouragement, but chess doesn't care. Even a single mistake spells disaster against a competent player. In an evenly matched game, well, there's nothing quite like it. Chess, at it's best, provides unparalleled tension. A close match, even among amateurs, can be a riveting experience for players and onlookers.
So the obvious way to enjoy chess is to find someone who is your equal. In a way, this is the golden age of amateur chess. There are several chess servers to connect to strangers around the world with comparable rankings. No need to register for tournaments or join a chess club. The Elo system long used by chess to rate players means you'll quickly be matched with competition at your level. There are even computer chess "players" who don't just play expertly, they play like real players, mistakes and all.
But what if you want to play in person with people you know? There are various handicap rules such as starting without one or both rooks. (Be sure to advance the pawn the rook would have defended so that you don't start with a hanging pawn!) Setting a timer for the better player can help too. But these are blunt instruments. What if the handicap isn't enough?
One solution is to move to the genre of chess puzzles. Again the depth of chess research helps because there are endless puzzles to try out out. (The books I recommended above include hundreds.) For new players (and especially children) this is a good way to practice without feeling discouraged. The more advanced player can set up the position that should lose and see if the other player can find the solution. Then play out the game from there or work together to solve the problem.
You might object that this isn't playing chess so much as teaching it. And you would be right. I could argue that the core of chess is the midgame maneuvering so skipping the opening is fine. So another alternative is to use the Chess960 varition, which starts the game with the back row in randomized positions. I haven't played this variation myself, however. Instead, I use mulligans to level the playing field.
Since the earliest recorded game, there's been a strict tradition of not allowing players to take back moves. It's a good rule for competitive chess, of course. I've often discovered a mistake moments after letting go of a piece. That said, letting the other player take back a mistake allows the game to remain interesting. If I feel the other player has made a mistake that costs them a causal game, I'm happy to let them take it back.
Ultimately chess remains interesting because a small number of rules produce a game balanced on a knife edge. Many outstanding minds and untold computing power have been unable to solve this puzzle. But that doesn't mean we have to be so unforgiving.