The sentence "I don't play golf and I never go fishing" can be translated into Spanish as follows: "No juego al golf y nunca voy a pescar".
As you see, you must use the present tense, which is one of the most used verb tenses (both in Spanish and in English). The present tense can be used in many different contexts, but here you use it to express the idea that an action -habit, hobby...- is not repeated or usual.
As far as your second -and related- question is concerned, "nunca” is the adverb of frequency used in Spanish to indicate ‘never’.
I'll add further detail as to how "nunca" translates into English: it may translate into "never" or "ever" depending on the structure of the sentence. In Spanish there is no grammar rule against a double negative in a sentence such as "No voy nunca al río", which would translate into "I never go to the river" or " I don't ever go to the river". In this case, the word order of the Spanish sentence is important: "No voy nunca al río" is OK, "Nunca voy al río" is also OK; but "No nunca voy al río" or "Voy nunca al río" are both wrong.
Yes, both of them are correct: "a pescar" and "de pesca". As far as the sentence is concerned, "no voy nunca de pesca" is as idiomatic as "nunca voy de pesca".
Optional answers to mmms1981 :
No juego al golf y no voy nunca a pescar.
No juego al golf y no voy nunca de pesca./ ... nunca voy de pesca.
I've seen both "ir de pesca" and "ir a pescar".
L no jugar al golf y yo nunca ir a pescar