Basically "grillen" means the same as "to have a barbeque", it's just a single word to express the same action. It's like if you would have a word "to barbeque" (a verb) in English.
Also in your example you have "barbeque" as a direct application to the predicate. I'm not sure, if "application" is the right grammatical term, see also adjunction, addition, append(ing) and so on.
— Jane reads a book
Jane - Subject
reads - Predicate
a book - direct application
— Tom runs accross the street
Tom - Subject
runs - Predicate
street - indirect application
So the right ways to build your example could be:
— Im Sommer grillen wir oft im Garten.
— Wir grillen oft im Garten im Sommer.
If you have a complex predicate like "gemacht haben" (machen, Perfect Tense), you should place the alterable part of the predicate on the second place, and the rest goes to the end of the sentance.
— Wir machen Fehler.
(We make mistakes)
— Wir haben Fehler gemacht.
(We have made mistakes)
And adverb is something completely different. Adverb works like a description of some feture of a subject.
— It was incredibly tasty!
It was tasty. How tasty was it? Incredibly.
— Es war unglaublich lecker!
Es war lecker. Wie lecker war es? Unglaublich.