Oh my - you're going to get a boatload of advice on this one, and most of it is all different. :-) So let me start the ball rolling by describing what works for us.
To start, I am 5'11" and 220 lbs, and my lovely stoker is 5'3" and 120 lbs. It's quite simple for me to hold us upright. What works for us is that she stays clipped in *always* unclipping only when she's getting off the bike. When starting out, I clip in with one foot, and backpedal until that foot is at the top of the stroke. Then I simply push off. We're both pushing down on that pedal, and we're off. It's a simple matter after that for me to clip in the other foot. The advantage to this scheme, for us, is that she is providing power right from the get go to get help get us moving quickly.
That's what works for us.
What works for you might partly depend on the disparity of weight between the two of you, and whether she's comfortable/trusting enough to stay clipped in all the time.
I know many, many couples who do it very different. For those teams, they both put one foot down, and when it's time to take off, they recite some variation on "one two three GO", and they're off.
At the end of the day, there simply is no 'right way', there's only the way that works for you.
But please trust me when I say that, with a little bit of practice and a little more time on the pedals, those wobbles WILL go away and you'll find yourselves riding together smoothly and effortlessly and marveling at the pleasures that only working together as a team on a tandem can bring.
Good luck!
ps - nice score on the Cannondale! Those are mighty fine bikes.