You must okay it with the landlord and I'd definitely get renters insurance if you don't have it already. I'd also get an electrician to look at the wiring and receptacles in that room. You'd do well to have GFCI outlets installed, as well....and plenty of them. You don't wont to overload any single outlet with too many things plugged into it.
Yeah, people who rent apartments don't start up fish rooms. Home owners start and maintain fish rooms. Adults with decades of experience start and maintain fish rooms (I'm sure there's some younger folks out there with fish rooms too). The first time your landlord sees more than a few tanks in your apartment he/she will consult your lease agreement and point out the breach.
You may want to reevaluate your plan. You're young and have plenty of time to have fish rooms in the future...