I'd say you're off to a good start with changing the oil and filter right off. Make sure the oil meets the Mopar specification for your particular engine.
The Renegade should give you a message when the oil needs to be replaced, depending on usage. Every 5K miles is on the too-frequent side.
And replace the engine air filter. Cabin air filter too.
I'd definitely replace the brake fluid. The requirement is every two years, no matter how many miles. If you don't know whether it's been replaced, replace it.
Beyond that, did you get an Owner's Manual with the Renegade? If not, manuals can be downloaded. There's an awful lot of good information in there. To include the maintenance schedule.
Replacing spark plugs may be in order, depending on your engine type. And I'd replace ignition coils at the same time.
Other than that, I'd get a good shop to do a thorough inspection of everything... Underbody, suspension, etc. They'll know what to do.
Special engine fluids? Like I said, Jeep recommends a specific oil specification, depending on engine type. Also coolant specification -- don't just blindly use generic green Prestone stuff.
What grade of gasoline to use depends on the engine. The Owner's Manual should tell you; I suspect Regular should be fine.
You mean idle for 10 minutes before driving? No.
Thanks for the advice! I had read on another forum that oil needs 10 minutes to heat up from a cold start, so it was recommended to let it idle before driving. Does that only apply in the winter, or not at all ever? Is synth oil recommended, or should I give it whatever the owner's manual suggests? I want to make this vehicle last for as many years as I can.