I like eggbeaters. They don't clog up as much as a set of Shimano pedals I had. You can find them online for around your pricerange. I've only used the Crank Bros and Shimano so maybe others differ but I doubt it. You press down and forward to engage and rotate your ankle to the outside to disengage. Each pedal comes with a cleat suitable for use in its makers line. All shimano cleats fit shimano pedals and all Crank Bros cleats fit .... Maybe Time or some of the others do it differently but again, I doubt it. The cleats don't work on different brands, as far as I know. Providing the pedals are the same the shoes can be used on any bike but MTN shoes are a little flexier than road shoes so you can hike if needed.
I strongly recomend some practice before hitting the trails full on. It takes a few rides to get used to the disengaging. Try playing on a grassy field. As you get better you will pop out with very little thought. When riding and reengaging a foot, just put your foot on the pedal and let the foot find it's way into the cleat. Looking down and fumbling for the cleat while trying to ride can be frustrating . Enjoy and have fun.