Second question first: "How much you need" is something only you can answer - and depends on what exactly what you are hoping to power, and for how long.
First question: Are there restrictions? Yes, there absolutely are. Unfortunately, the limits may not be enforced consistently everywhere. In the USA, the FAA-imposed limit on batteries (that's what a "powerbank" is) rated at 100 watt hours or less (over 100 Wh would require prior approval from your airline, which is a hassle to get and while theoretically possible, no way I would bother). The good news is that 100 Wh is a pretty big one (but there are larger sizes). Yours should have its rating written or stamped or etched on the outside. I've got one here that's rated at 96 Wh and 26800 mAh (just barely below the limit) and it's a beast - measures about 7" long X 3" wide X 3/4" thick. I've used it to power several phones and tablets on world-spanning trips and it carries a lot of juice (about $66 on Amazon - here). Great unit. Note: all of these things are surprisingly heavy (batteries, you know?), so don't go overboard, you will need to carry them. You'll also need a charger block (wall-wart) and cables.
Other countries have different limits. My understanding is that China has a lower limit and is also among the most strict and intrusive when it comes to enforcement.
For most consumer powerbanks (small-ish sizes) you will be fine. You will typically need to pull it out when going through airport security inspections (considered "electronics").
Also keep in mind that USB power ports are increasingly common in airports, on planes and elsewhere, so you may not really need mobile power sources as much as you think. And then there's the notion that you are going there to actually experience the place, so it may not be a bad thing that lack of power might cause someone to stop looking at a screen and look up at the real world for a while (there have been times when this was actually a blessing, though the owner of the screen didn't feel the same way at the time).