To picnic is a recognized verb.

You can say go picnicking. It has the same meaning as going on a picnic or having a picnic. Note that some collocations may be more or less common in certain dialects - like some people say have a shower, while others say take a shower.

When you're inviting someone to go with you on an activity, you can substitute go with come. So, you could say come picnicking with me, or come on a picnic with me. To me, this sounds friendlier than go. Come with me emphasizes that it is with the speaker. Go with me emphasises the place you are going to.

Hi Ada,

A picnic is a noun; picnic is not really used as a verb or gerund in a phrase like go Xing. You can say “Come have a picnic with me”. If you want to mention a specific location you can use in X town/city, at X park/beach. At is more common for locations within cities or towns.

To picnic is a verb.

"Go on a picnic with me."

Example of location:

"Go on a picnic with me in Texas."

"In" is used for location; "At" is used for time.

Example of time:

"Go on a picnic with me at 11 A.M."