no encounter cards stay out, but some mythos cards do. Like rumors. Generally they tell you when they need to stay out usually with some counters. I suggest if you haven't already playthrough it yourself first with 1 or 2 investigators to get the hang of it. My brain melted the first time I read the rule book, but as I played it all started to click. What I generally tell people that get intimidated the first time I bring it out. Is I know this looks intimidating, but all the game really boils down to is pull a card, do what it says and make a roll. Thats the game. The cards generally tell you exactly what to do. So you follow instructions and roll. Rahdo has a run through as well
https://www.youtube.com/watch?v=_iLIJ-DXiQ0 . Sometimes you are kinda stuck in the game with the mysteries, since it might be a clue related one, and their aren't enough clues or gates to complete it. Particularly in the beginning of the game its likely the first mystery you just won't be able to do anything about for the first few turns or so. So in the beginning make yourself stronger. Go to locations that are likely to improve a weak stat, buy items to help you, just power up in general. Who you are playing as matters too. My advice unless its a big group of people, don't play anyone with awful influence,(its used too often) and at least have 1 person set to take out monsters. There are many great characters my favorite is trish since, she can get free clues for rerolls, and she starts with the .45 which immediately makes all her attack rolls with 6 dice.
I've played it about 30 times or so, I can try and help with any rules questions. Honestly I think the big thing about arkham horror is with its expansions it probably becomes convoluted. The base game isn't that different from eldritch horror, except there are a few more things to keep track of that don't activate every time, that you have to remember.