Once you have registered on the Surfshark website you can use the user and password to operate on your PC or mobile phone but not on WireGuard, for that you have to use the private key, there is no way to use user and password that I'm aware of.
To obtain the private key you must install the application for Windows (I'm talking about Windows because I use that OS but I imagine there is an equivalent for Apple or another OS).
Once logged in the app choose "VPN/Manual configuration" in the left column, In the second column choose "Router" and in the third choose "I don't have a pair of keys" or something similar, I don't know how it is translated into your language.
From then on, follow the on screen instructions and when you have obtained them remember to write them down because if you lose 'em you cannot recover it, you will have to make new ones.
Anyway, in the application itself there are plenty of instructions.