Which sentence is correct and why:

After we leave there, we will do the following: get a map, eat lunch, and explore the city.

or

After we have left there, we will do the following: get a map, eat lunch, and explore the city.

Thanks for your help in advance.

The future perfect tense is used for an action that will be completed sometime in the future before another action. So, your "After we have left there" is fine.
"After we leave there" is simple present tense. It works fine too for the future in adverbial clauses. It would be a strict grammarian who picks you up for this.