I would hope you inspected the roof, caulking etc before you purchased. With a purchase this old, you might have been well served to pay a professional to inspect it. But, if that's water under the bridge, my first checks would be roof and caulking etc.. make sure it is waterproof. Then, wheel bearings, brakes, suspension inspection, tires.. as pointed out, if they are more than 5 years old (I say 3 personally), then replace them. Trailers do not move as much as cars, so dry rot is inevitable.
Track down all the grounding locations, and make sure there's a good ground. Remove, reset as necessary. Ground problems are a major cause of electrical gremlins.
Consider taking the battery to your local autozone/advance auto parts for a battery check. It's free.
You are on the way to a new hobby and great times for you and your wife. Enjoy it all.. even the maintenance and check out. You are getting to know the RV better and that's important when you are out on the road and something happens.
Mike