All three of your sentences are correct. And this is a very important and tricky topic in English. There are almost innumerable verb clauses that use "go". Examples: to go crazy, to go postal, to go back, to go under, to go around, to go through, and to go down. If a destination is the subject of a sentence with the word "go," you will need to use the preposition "to" (example: She is going to Albany for Christmas. He is going to college next fall.)

All three are correct. 1) and 3) are similar. "Let's go + verb". 2) "Let's go + gerund/verbal noun". Let's go get some drinks. Let's go play basketball. Let's go fish. Let's go eat dinner. Let's go shopping. Let's go fishing. Let's go swimming.