Getting reputation for the new factions is going to be beneficial in different ways. Earning reputation towards the factions is going to help you unlock Pathfinder in Kul'Tiras and Zandalari. Pathfinder is nothing new so i decided to include it in here. Pathfinder will give your character increased mount speed and its account wide. Also gaining reputation will allow your character to buy gear, from a vendor, equal to Heroic dungeon quality.
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As late as possible. Early on, you’ll want to follow the missions in Battle for Azeroth right up until you get the option to send a follower on a two-hour long quest. As soon as you get there, stop doing the War Campaign missions. You can come back to them once you have reached something like 119 and a half and finish leveling with those if you want. If you don’t use these missions to cap off your leveling journey, then they’re the first thing you’ll want to do once you reach max level.
I found that in most cases, I could pick the zones I liked and skip everything else. It was slower, but I still wasn't forced to do quest zones I hated (and, notably, I skipped the entire Cataclysm expansion's content -- sorry, Thrall.) I did do dungeons, both in group finder and by two-boxing (more on that in a moment as well), but only once for the quests, and mostly to clear what felt like an awfully long slog from 40 to level 80.
Something to keep in mind as well is that this will make enchanting materials more valuable, since it is likely most players will now simply scrap their boe greens instead of getting them disenchanted. Also redundant dungeon/raid gear can be scrapped rather than traded to the (guild) enchanter in the run. Less stuff gets disenchanted, meaning there will be less ecnhanting materials, meaning their value will rise. All the more reason to DE rather than scrap as an ecnhanter.