Board Game Arena

I am sorry but navigating the website to find this info is not the fastest thing in the world, I chose to ask here instead. I am thinking to try some more turn-based matches because it is hard to find opponents, especially in the Arena, in certain games. But what is the limit for free members, how many different games can you be in at any given time? Does it affect your ability to play real-time games also?

As far as I know, the number of turn based games you can play at the same time has nothing to do with premium or free but with your karma and how many games you successfully completed in the past.
So in the beginning you can play 4 or 5 (I think) turn based games at the same time, and after you have finished your first game, the number goes up. There are people with hundreds of simultaneous games.
Edit: real time games are not effected by your turn based games unless your running over time, in that case you might have to make a move in your turn based games first, before you can start a new game.

Thanks, it seems people play more turn-based ones in general, so if I want to play I will have to switch to them, at least for certain games that last longer.

Yes, I think for a lot of people it is easier to play a couple of minutes throughout the day instead of 45 minutes at a time and since you can have several turn based games at the same time, there should be more of them than real time games (at least for longer games).
One piece of advice if you are starting out with turn based games:
Make sure you look at the playing hours for fast turn based games (and maybe turn the flexibility off). It is very easy to get placed into games that have playing hours when you are sleeping when you are using the automatic match making.

Thanks, I will be careful, although I am not sure I want to get into the turn-based format for now. I am trying to find more real-time games that actually have people playing, not always easy though.

Are the "playing hours" I am shown in my own timezone?

Yes