Learn Finnish: Lähdettekö te viikonloppuna juhlimaan vai kalastamaan? - Are you going out to party or to fish this weekend?

They are in the third infinitive form. The main verb in this sentence is lähteä (to leave, to go out). When you use lähteä to say you are going somewhere to do something, the action verb must take the -maan or -mään ending.

How do we build juhlimaan and kalastamaan from their dictionary forms (juhlia and kalastaa)?

You find the strong stem of the verb, which is the same stem used for the "he/she" form (hän juhlii, hän kalastaa). Then you just add -maan. Because neither word contains the front vowels ä, ö, or y, they both take -maan instead of -mään.

Why is there a -na ending on viikonloppuna?

That is the essive case ending. While this case often translates to "as a [role]", it is also the standard way to express certain points in time, like days of the week. Here, it translates simply to "on the weekend".

Could I use tai instead of vai for "or" in this sentence?

No. In Finnish, vai is used specifically in questions when you are offering a direct choice and expect the listener to pick one. Tai is used for "or" in normal statements.