Dnd 5e // I am missing lots of stuff in the Compendium

Well, as said, I am missing the additional Sorcerer Origins (Wild Magic, ...), Druid Circles (Circle of the Moon, ...), Warlock Patrons (Archfey, ...) and other backgrounds than Acolyte. Are there particular Compendium Expansions that are needed to be bought? Which ones? Volo's Guide just gives additional Monsters and Races.

The Compendium only provides options published in the SRD, the freely available System Resource Document. It only includes 1 feat, for instance, a slightly limited selection of spells, and 1 subclass per class. At this point in time, Roll20 does not have the rights to the Players Handbook.

So I need to look for buying a physical book. Which one's would I need for a complete set of player options?

The Player's Handbook is the main book you need for player options. There are additional books you can purchase for extra options on top but these aren't necessary to play: Sword Coast Adventurer's Guide Xanathar's Guide to Everything None of these are currently available through roll20. The dev team are working on changes to their backend to possibly accommodate these books' content at some point in the future, but there is no known eta at this time.

Another major expansion is Volo's Guide, which adds half a dozen new races for players to use. There is some overlap between the expansions.

You can shearch on tab table