Get comfortable on the bike first.
I would spend a week or 2 getting comfortable on the bike in general if you are not accustomed to riding this type of bike.
I would have a LBS check the tension on the pedals and set them to the losest setting.
When you clip out, are you clipping out on only one foot to set it down? That's how most do it. When I go for a ride my left foot clips in at the beginning (before I ever start peddaling) and i stays clipped in until I get off the bike. (Don't make the same mistake I did and learn to clip out with the right foot, it makes chain ring tattoos on me all the time! Clip out on the left while you are learning so it becomes second nature). It may take some practice for balancing the lean. Do this in your non-clip shoes. Try leaving the right foot on the pedal, and only putting your left foot down. Gently lean a LITTLE to the left as you do so that the bike will lean in the direction of your free foot.
Soon enough it will be in your muscle memory to lean the right way, which will avoud toppling over toward the clipped in foot.
I can't say I recommend this BUT when I was learning, I didn't have anyone to teach me the "right" way. So I learned to clip out by actually rotating my foot TOWARD the bike, rather than outward. For some reason I was stronger in that motion. Not sure this works with your type of pedals, but on SPD's it does. I try to swing the foot outward to clip out now, but if I'm in a hurry to unclip, muscle memory takes over, and I swing toward the bike and pop right out.
It sounds like you are fine on the trainer, so you need to practice the leaning of the bike with one foot still on the pedals.
When I come to a stop, I unclip the foot and hang it downward off the pedal. I squeeze the brakes to time my stop to wear my clipped in foot will stop at the bottom of the pedal stroke. As I'm slowing and my clipped in foot is rotating downward I stand up a little, clearing my rear end from the saddle so that I land my clipped out foot down on the ground and I'm standing over the bike, but in front of the saddle. I can not place a foot flat on the ground while still in the saddle. If I tried to, I'm pretty sure I'd fall over.
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one other tip for clipless pedals
I second or third the recommendation to pick one leg/side as your clip out side. This helps you remember to always LEAN in the same direction too.
One of my first falls was coming to a stop in front of a group of experienced cycling friends and saying 'look at me I'm clipless', then promptly leaning in the direction I had NOT clipped out of. ooops.
Besides picking the side, the other thing I tell my trainees when they are first starting out is, if you even think you might need to stop for any reason, or it starts to cross your mind to touch your brakes, clip out. If you're in traffic and approaching a light you don't know the timing of, if you're on a bike path and coming up on some unpredictable peds, if a rider in front of you is being squirrelly, etc etc, clip out. and yes yes yes to loose tension.
I'm a big fan of Speedplay pedals. You get a little side to side motion before you disconnect which is great for folks with knee issues. For some folks the lack of play in other pedal systems causes problems. The only downside is that the pedals are very small, they are just a small lollipop sort of thing, and you may be tentative about standing up on them initially when climbing, but once you get used to them and their lightness and ease of use, you may just be as hooked as I.
There was a great article in Bicycling Magazine about clipless pedals. I'm going to try to attach it here, not sure if it will work. It was just a few months ago. I found it really interesting and helpful AND persuasive. ;)
