I know that Northern Cyprus is recognized only by Turkey, and not other countries.
So how can I visit this republic - only from Turkey, or there are some other ways?
Will I have problems with getting Cyprus visa after such trip?
It seems to depend on what passport you have. Both the tourist information website and the Ecran Airport site say the same thing. For tourist trips of up to 90 days, citiziens of European Union (EU) nations, Turkey, USA, Canada, Mexico, Israel, Japan, Singapore, Switzerland, Iceland, Norway, Australia, New Zealand do not require visa.
It looks like all other countries do require visas, you'll need to contact your nearest North Cyprus Embassy to get one. And if there isn't one, then it looks like Northern Cypriate matters will likely be handled by the local Turkish Embassy, so speak to them.
As for issues with the two Cypruses, if you see the "Crossing the Green Line" section on the tourist information site it suggests you shouldn't have any issues as the land border is open. If you've not got a suitable EU passport or visa, you'll need to get a visa to cross to the south, and you can certainly ask about it when applying for that visa. I wouldn't expect any issues though.