"I like to go cycling and play chess."
"I like to go cycling and I like to play chess."
"I like to go cycling and to play chess."
These are all acceptable but they can have slightly different nuances of meaning. The first one might give the impression that you are going cycling in order to play chess or as a prelude to playing chess. For instance, you may enjoy cycling to the local chess club meeting.
It can even suggest that you play chess while you're cycling! Although that would be an unlikely interpretation in the context of this example, it would be less unlikely if you were saying, "I like to sing and dance." There's certainly an indication in this construction that the two pleasures may be connected in some way.
The second one suggests most clearly that cycling and playing chess are two distinct activities which you can enjoy independently of each other. ("I like to sing and I like to dance.")
The third example is a shade more ambiguous although, like the others, its grammar is perfectly sound. It's not clear whether or not there's any connection between the activities (other than the fact that you like them) when you use this construction. The most likely meaning relies more on context than grammar.
Sometimes writers get bogged down by questions of grammar or style when the real problem that's worrying them is clarity of meaning. This can often be easily solved by changing the construction of the sentence altogether. E.g., "Chess and cycling are activities I enjoy." Or even just, "I like cycling and chess."