Always best to ride with a buddy. Go with the neighbor even if that horse doesn't like to lead, it's good for them both. Ask around, join a trailriding club if there is one locally. If you must go down the road, stay on the unpaved area as much as possible - riding with shoes on pavement is like iceskating on horseback. If they slip, they really slip. I like the suggestions of what to bring, but I always like to ride on trails with the halter on under the bridle, and with a lead attached. It's a good back up if one of the horses balks at something - the other can pony them on through if they refuse to just follow, and if you need a break along the trail, you have a way to tie them.
Take a small bag, purse, or saddle bag to keep a bottle of water and extra fly spray in, and keep a cell on you (not on the horse, in your pocket). That's about all. Have fun. It's good to change up the horse's routine.
Ask Often, Demand Nothing, and Reward Generously.