I would like to upgrade my Shimano disc rotors from 160mm to 203mm. How to pick the right rotors? My bicycle came with the BR-MT200 (https://www.bike-discount.de/en/buy/shimano-br-mt200-disc-brake-set-721603) brake set: hydraulic levers, 160mm rotors.

I'd like to upgrade to 203mm, but I don't know what rotor to choose. My curent ones are marked SM-RT10-8.

How do I pick compatible ones? All I know is that they are centre lock. And should I be upgrading both rear and front, or front only?
To upgrade rotors you'll need to change the disc brake caliper mount adaptor too. This is specific to your fork but as you currently have 160mm disc, you'll need to an adaptor to move your caliper out for the 203mm rotor to fit. See this post for a little on that. 

For what disc to get, it all depends on how much money you want to spend. If you're just using the bike for commuting, just get the cheapest shimano disc you can find, if you want it for mountain biking, are worried of disc heating up and can afford it, then get a more expensive one with cooling fins and specific metals for that. 

Depending on what you do with the bike is what should determine the size of the rotors, if your rear disc is working fine and doing it's job, there is no need to get a biggest rotor. If you want more stopping power or are having problem with them overheating, then get a bigger rotor on the rear. 

There is nothing to say you shouldn't have different size rotors front and rear. 

When you're getting a new disc though, you need to check the disc width is compatible with your brake. If you have a Shimano brake, then getting a shimano disc makes sense. If you want a diffent brand disc, double check it's Shimano compatible before buying.