This is a mortise lockset (meaning the mechanism fits into a pocket carved into the edge of the door) with deadbolt. If you just wanted to rekey it, the cylinder (the portion that recognizes the key) screws in from the face of the door and is kept from unscrewing by a setscrew. That setscrew is accessed by removing the faceplate from the door's edge.
A locksmith can rekey or replace the cylinder, fairly inexpensively, to change which key it responds to. (A few minutes of work if you have the right knowledge, tools, and parts.) Even a large home center is likely to have replacement cylinders, though getting one with the right tailpiece, or moving the tailpiece across, may not be obvious to a beginner.
However: YES, you need to get landlord's permission. Your lease undoubtedly has a clause regarding unauthorized alterations to the rental unit, and even if you don't have a formal lease a court is likely to side with them if they take the cost of regaining access out of your security deposit. If you think their having access now is bad, just wait until you have given them reason to distrust you.
Possible but unwise without permission, and the only way you're going to get permission is if the landlord get a copy of the new key. Which might still be worth talking to them about if you think there are copies of the key in the hands of past tenants it their friends, which is probably a bigger risk.
(ORK Security Services; full-service EXTREMELY part-time locksmith, basically inactive except occasional jobs for friends.)