I definitely think it's a hard one to prepare for. I'm running it at level 6 right now, on my 6th or 7th session. My advice to you is to steer your PC's towards the more fun parts, so you don't have to plan for everything.
Adventure Hooks
My players come out of the Lost Mines of Phandelver, so they are already in factions, and just got the faction hooks. A factionless player got a Windharrow adventure hook. I would simply designate adventure hooks, or speak with the players about it. Most players will forget any information from these hooks, if you keep it vague.
Initial play
Initially, the Feathergale Spire is both the most obvious first location and vastly the most interesting of the elemental cult hide-outs, from an RP standpoint. The manticore hunt is exciting (albeit not very challenging) if done right. You can easily make the Feathergale Spire a non-combat (besides the manticore hunt) encounter, that ends with the party being flown to the Sacred Monastery (although the combat here is not that interesting, it does have a lot of information, and the most obvious way into the depths below).
I have not run the fire elemental camp, but it also looks interesting, both from an RP and combat standpoint. The Monastery and Rivergard Keep are not too interesting, in my humble opinion. Be aware though, that the fire elemental camp is pretty high-level.
Random Encounters
I suck at improvising (or at least, I'm not that good at it) so I actually sat down and fleshed out some random encounters. I did away with some I thought were uninteresting, and improved upon others with stuff I knew my players would like. I still roll for random encounters, but from a smaller, and more prepared list.
Overland travel
The map is very, very large, if you follow the guidelines on the map. Another thread on this forum has compared this map to other maps, and scaled it down a lot, which reduces travel times heavily. I advise you to do this, and read the entirety of that thread, since it really makes travelling a lot less tedious for your players.
I've personally introduced a random weather table that I roll on, to clue my players in that weather is behaving very weirdly. It doesn't have to be anything fancy with mechanics - it could be as simple as opening the DMG and choosing a type of weather for the day.
Read others experiences!
My small post here is a lot less than you need. I would advise you to look around here:
https://www.reddit.com/r/dndnext/comments/36vwdy/princes_of_the_apocalypse_resources/. It has everything you need to be well-prepared for running PotA (which, sadly, is a lot more than should be necessary when running and out-of-the-book adventure.
My last advice, that I try to follow myself, is to remember to have fun with it and be creative. If you suddenly change a name here, or give out some information that doesn't match with the book, just go with your own direction. The book is meant as helpful guidelines, the adventure is yours!