You're opening up a huge can of worms asking for string advice. My two cents is to start with a set of 12-52 round wounds. It is my opinion that one should use the heaviest string that you are comfortable with. 12-52 is a good starting point. If they're too heavy for you, then fine, go with a lighter set. But at least give them a chance. And it will depend on the guitar. I have 12-52 on my Telecaster, 11-49 on my Strat, and 10-46 on my Les Paul. I've got 13s on a couple hollow bodies so it can go that way too.
Lighter strings are fine but if I can avoid tuning problems and just get an overall bigger sound by going heavier, I will. Especially if my fingers can handle it. Again, depends on the guitar. But 12-52 is a good place to start.
As to flatwounds, I have used them for years. They feel great, they're quiet, there's not THAT much difference in tone....but I feel like I'm getting more tactile feedback with roundwounds. It's tempting to go with them, and I have, and I probably will, but then I find myself almost bored. I go back to squeaky roundwounds and it just feels like I am more in control if that makes sense.