1. They get out of the city and spend time in nature.

2. They go out of the city and spend time in nature.

(Are both the same in meaning? Which one do you use more, go or get?)

3. Everyone will have fun there.
4. Every will have a good time there.
5. Everyone will enjoy himself there.
6. Everyone will enjoy themselves there.
7. Everyone will enjoy himself or herself there.
8. Everyone will enjoy himself and herself there.

(Are they all grammatical? Which ones are not the same as #3?)

1 and 2 are fine and mean about the same thing. I'd use 1 more than 2.

These are fine: 3, 4, 7

These are either incorrect or never used: 5, 6, 8