What's the fastest way to progress through the game?

Run while you still can!
Joking aside, events are probably your priority in practically every circumstances because they give out rare ascension mats that are often a pain or practically impossible to get in normal circumstances through farming. Fgo is a grind fest so you will need to collect as much material as possible to improve your team. Sometimes event also give out holy grails which are limited time only so you should try to get those. Since you're a new player, start by gathering resources and leveling up your servants as you will need them to clear tough challenges. Gachas are bad civilisation and you don't actually need them to beat hard content in this game as the same can be done using 1 to 3 star servants (notably hans and mash), so roll for who you like :). Try your hand at the event would be my recommendation. Crystalized lores are also event exclusive and are the only items that can get a skill from level 9 to 10. They are invaluable in the long run as they give that little 1 turn CD and boost at the end.