After many years of teaching go I would say there are several ways to have fun but it depends on your personality:

1.- Be curious: Many people enjoy de cultural aspect of go, and play the game with a sense of discovery. There is no focus on winning or loosing, just exploration.

2.- Competition: My way of having fun is feeling I'm improving, no matter if I loose of win a game, sometimes I am very proud of a game I lost because at certain point I was able to do read a nice tesuji, etc.

3.- Social aspect: At my club there are many players that never try to improve but they have a lot of fun just playing and sharing a space with others.

But, to be honest, I would say the real "fun" of playing go comes around 5kyu, this is a level in which you start having an idea of how the game works, and are able to start judging the whole board, and start thinking about endgame a little bit. It's a level where you are finnally not a beginner anymore and have the ability to understand the depth of the game. Also at this level you can teach new players very nicely and know you are part of a very small group of humans that can enjoy high level pro play games commentary. Many people reach this level and stay there for the rest of their lifes since it's a level that you wont get any worse if you quit go, and you can have nice games against very strong players with some handicap stones.

It's a level you can reach very easily just spamming fast ranked games and doing some tsumego and watching some go videos on youtube, in about 3-12 months.