Quick answer
На приро́ду (accusative) for going out there, на приро́де (prepositional) for being there. The usual motion-versus-location pair.
The kit
The fire and the food
The place
On a trip
Шашлы́к is the point
For most Russians, going на приро́ду means marinating meat the night before, driving somewhere with a river, and spending the afternoon around a манга́л. The invitation Пое́дем на шашлыки́? means the whole outing, not just the food.
Fires and rules
Open fires are restricted in dry summers and banned outright in some regions and periods — the term is противопожа́рный режи́м. A манга́л is usually still allowed where a костёр is not. And leaving the site clean is a point of pride: убра́ть за собо́й — clean up after yourself.
Typical mistakes
Мы е́дем на приро́де.
Мы е́дем на приро́ду.
Motion towards takes the accusative.
оди́н дрова́
дрова́
The word exists only in the plural.
Check yourself
Which is correct for “we’re going to the countryside”?