Welcome to Early Access, where features are missing everything is buggy. Zombies flat out don't work right now. There are barely and of them around, they move weirdly and walk straight trough fences, walls and entire buildings. There are no vehicles yet, weapon variety is low and there are many health conditions still missing.
I too have had a lot of downtime in that game, because stuff does not happen unless you very specifically make it. Survival is far too easy when the zombies are so bad and so many illnesses are missing, so the only real threat is other players. If you want action, stay in the towns and cities around spawn, or better yet go to the north-estern airfield. Always something going on up there.
I'd recommend you not play the game right now unless you are very specifically interested in doing so right away, damned be the bugs, glitches and unfinished features. I only play it very occasionally, and when I do it is with friends. And we don't take much care surviving, we're more into trying to get ourself into sticky and entertaining situations with other survivors!
...and of course it has no objectives/quests. It's working by the same logic as minecraft: you make your OWN objectives, quests and stories! You can even write in the game now with pen and paper, I'm doing my own little diary for people to find when they kill me.