Just came back from another ride. This is my third day and it's beginning to feel a better. My rear end only feels uncomfortable on the uphill, so I up my ride to 2.2 miles. For most part it only gets wobbly when I try to shift. There are a couple of things I noticed:

1. I seem to "use my hands" too much. When I'm tired and not paying attention, my left hand tend to lean on the handle just hair more and it threw me into a wobble.
2. I feel a lot more stable if I consciously keep my knees tight (it also seems to make it easier to stay on the pedals).

As someone suggested above, I tried to "feel the bike beneath" by gently changing the balance back and forth using only my lower body. This is definitely a Aha! moment: if I can train myself to do so consistently, I won't have to rely on my hands to rebalance (which is likely what caused the wobble to begin with).

About not thinking too much, that's definitely one of my problems . What I've been trying is to do all my thinking before I get on the bike, and pretty much let it roll once I started riding. The only things I consciously think of during the ride is when to brake, when to shift, keep a good pace, etc.

I have a question about riding position. Someone on the board suggested I should learn to bike in lower gear, pedal faster but go slower. Before I bought the bike, I read through pretty much all of Sheldon Brown's articles. One thing mentioned in the saddle piece is that the saddle isn't meant to hold the entire body weight, and much of the weight should be on the pedal instead. It seems to me that in order to pedal faster, pretty much all my weight has to be on the saddle and use the handle bar as a support. Otherwise, I feel that my legs can't spin fast enough to keep up with the low gear. What is the right amount of weight on the saddle?