Congrats on your new (to you) trailer!
Just off the top of my head I would:
Check the date code on the tires. If they're more than 5 years old it's probably time to replace.
Check the seals on the roof and around all of the windows and seams. Recaulk as needed.
Service the wheel bearings.
Check the refrigerator & water heater burners to make sure they're clean & insect free.
Check the anode in the water heater.
Go online and download as many of the manuals as you can for the fridge, heater, water heater, awning, etc. You should be able to find most of them and they will list the required maintenance. Plus they will have operating and troubleshooting instructions which will be very handy to have.
As for the tanks. Fresh water is the easy one there should be a simple stopcock type of drain on the bottom you can open and just drain it onto the ground. After that I would flush it really well and disinfect it. Who knows how long water has been in there and what's been growing in it.
The black & gray tanks will need to be cleaned also after you dump them. There will likely be 'crud' stucks to the sides that may give you a false level reading so you'll need to clean that as well. One trick is to dump a bunch of ice in and tow the trailer around so the ice scrubs the tank inside.
I'm sure there's things I haven't thought of so hopefully others will chime in.
2011 Chevy Silverado Crew Cab 4X4
2009 210 WBS