(season 5 PTR) Seems like I had all the pieces and the baby Groot-ish pet by the time I completed the chapter 4 seasonal journey stuff. Chapters 1-4 consist of real basic challenges that you would normally be doing anyway (running bounties, killing chapter bosses, enchant an item, equip all level 70 items, etc). I did all of it from adventure mode.
For the later tiers you get things like the seasonal picture frames, extra stash space... and you can earn the new wings for mastering the set dungeons.
Edit: Avoid confusion: The set dungeons are available on non-season too.
Actually, I don't think that doing set dungeons - you need to complete one to get chapter IV - really counts as "basic stuff". Set dungeons are an extra part of the game that you don't necessary have to visit to progress. And, pennants/wings for completing set dungeons aren't tied to season, as far as I know.
Back to the original topic: are we really sure that set items will be gifted in live version of S5? I personally think it was a PTR-only thing to speed up testing process.