Okay, most of these suggestions are great, but I'd like to break it down a bit for you if you'll bear with me.
I started getting a lot of break aways when I started playing rec hockey. But I wasn't scoring much. So I decided I would practice one basic move and perfect it before moving on. A couple of posters mentioned this, I'd just like to add some detail.
You can do this with a stick and ball or stick and puck if you have regular access to ice time. But I started with a stick and floor hockey puck. Practice makes perfect, btw.
First: Go in on net and just shoot the puck to the corner of the net on your forehand. What you also need to do is be very aware of your body position as you do this. Most people drop their shoulder a bit and transfer their weight to their back foot. While you are monitoring your body position, make sure you look at the exact spot of the net you are shooting at. Over and over.
This is a simple stuff, but the main thing to practice is the awareness of your body position as you shoot from the forehand...as well as looking at that empty space of the net.
Second, once you have practiced this...and become very aware of your body position, now you fake the shot and move it to the backhand. Faking the shot should still have you drop the shoulder, tranfer weight to your back foot and eye that empty part of the net you are faking your shot on the forehand. This is all designed to freeze the goalie.
And then you move the puck to your back hand.
Third, when you make the move to your backhand, make sure you bring the puck completely to the other side of the net and tuck the puck in that opposite corner. And keep practicing this. After a while start putting the puck up top off the backhand. After a certain level of competence, you may experiment with the idea that the goalie will make the big move to come across. As you get proficient, sometimes slide the puck along the ice into the net somewhere around the middle. This is where the goalie has opened up the 5 hole in the big move to come across.
Fourth, if you practice practice practice you will relax doing this move and start to see the open spots. Never look at the goalie's pads, always look for the spaces of open net. They are there.
Fifth...as you become really good (and relaxed) add in one final move. From the backhand bring the puck all the way back to the forehand once the goalie has made the big slide to stop your backhand goal. The goalie slides across, you have tranferred the puck to the forehand and he is out fetching hot dogs.
Practice practice practice is the key until your body remembers the move and you don't have to think much. Finally, as you begin to test this out in game situations, always decide on your move before you get in. Changing your mind in the midst of a move usually cause the puck to dribble off your stick.
Off this basic stuff you have a number of options. Shoot from the forehand to the corner of the net. Move to Backhand to the far side, on the ice. Backhand to the middle, along the ice. Backhand to the far side, up top. And backhand to forehand either up top or if there's room, along the ice.
Sorry for being long winded, but once I did this I started scoring on most of my breakaways. And I continued doing this playing old timers hockey. I was relaxed and knew what I was going to do off the options I had. Once I got better at it, I could relax to the point where I felt I could manipulate the goaltender.
Sorry for being long winded...but goal scoring is just so damned much fun.