Congrats!
Take it easy on those hills. If they are steep, up and down mowing is the safest way. If you ever feel tippy, turn downhill into the tip, not uphill.
A lot of people swear by putting Stabil in their gas cans every time you buy gas. The green marine Stabil is thought to be better than the standard red Stabil. I've been using it and I'm not having problems. I would still recommend running the gas tank dry if the mower is going to sit unused, like when you put it away in the fall. I dump gas that is getting old into the car and get fresh gas for the mower, generator, etc.