First thing I always do is try to expand towards my neighbors within reason, especially if there's a valuable choke point between us. I'll usually leave the planets that are behind me for later, unless they're really easy to capture.
I always do everything I can to start out as fast as possible too. I buy 100 crystal for early civilian infrastructure upgrade on my homeworld, scuttle one of my frigate factories (and sometimes my capital ship factory too), always build a carrier as my first cap ship, and pop out a migrator as soon as my 2nd light frigate is complete. I don't know how to start expanding any faster than that unless I built a colony cap ship. I'm usually playing on hard or unfair btw, and I can beat unfair on 1v1 usually.