I want to go out and have some fun.

Literal

Play-[purpose-に] go-want.

Minimal form — no subject, no politeness marker. 遊びに行く ('go to play / go have fun') is the outbound counterpart of 遊びに来る. The plain-form 行きたい is an expression of personal desire that wouldn't be used directly with a stranger but is natural to say to oneself or to a close friend.