Want to go on a picnic with me?

Literal

I [with-と] together picnic [to-に] go [question-ないか].

Invitation pattern with ~ないか. ピクニックに行く uses に for the destination/purpose — 'go on/to a picnic.' The informal いっしょに is written in hiragana here (one of several acceptable writings). A friendly, casual proposal.