Now, I'm not a fantastic pianist, but I've been learning for a few years now. While I'm not going to teach you to play, (there are plenty of other resources on the internet for that) I can share a few things that I've learned about playing. They all tie in to one golden rule: with practice, skill comes automatically. These principles can apply to practicing with any musical instrument, by the way.
Forget the tempo: go for accuracy - The most important thing to remember when practicing with an instrument is to slow down. Play as slow as you need in order to play accurately. With practice, speed will come automatically. You never need to worry about tempo. The correct tempo will come on its own. Remember that: ACCURACY is the most important part of practicing and playing.
No problem is insurmountable - As you're playing, you'll come across parts in a song that seem too complicated for you to play. Most likely, this is because you're going too fast (see above). There's no leap in skill or ability required to nail a particularly hard part, just keep practicing it over and over very slowly and the skill needed to play the part will come automatically.
Reading music will get easier - Reading music was always a chore for me at first. When I would get to a new note, I would have to stop, look closely at where the note was on the staff, then go over the acronyms in my head (e.g. FACE) to figure out what note to play. It's painfully slow in the beginning and really frustrating when all you want to do is play, but after several songs, you'll just start to remember what notes are where, and the link between the musical staff, the names of notes, and the notes on your instrument will strengthen to a point where, once again, it becomes automatic.
Take frequent breaks - As soon as you start feeling clumsy, or if you're like me, clumsier than usual, take a quick break. Do something else with your hands, like play a song you already know well or play a quick game of Tetris for a few minutes, then go back to practicing. This works wonders for me. Roughly for every 20 to 30 minutes of playing I'll take a 5 minute break. I'm convinced that I'm getting better faster simply because of this.
Okay, so that last one doesn't tie into my golden rule so well, but it is important, so I'm keeping it in there. Deal with it. Remember these principles and practice every day and you will be able to play anything. I promise.