I also use my dominant hand (right) only. That leaves my left hand on the front brake lever, although I generally don't brake and shift simultaneously.

Typically if I'm doing a double shift (front and rear) I'll shift the chainring first, then the rear. I'd be hard pressed to explain why I do it in that order, and I don't know if that's what's recommended. It's really a pretty seamless transition between shifts.

Depending on your gear, you might find that the smoothest shifts are obtained by overshifting slightly, then pulling back on the lever, especially when you up-shift the rear. This will become second nature once you've ridden a lot. Using a modern ramped freewheel will mitigate much of the need for this practice, but some of us like the old school clunk.