Sounds like you're off to a good start. Generally when I start playing the first thing I do is build one harvester for every planet in my home solar system and get them started with transferring resources to my home world. From there I build some probes and set them to exploring the galaxy. While I'm waiting on those guys to build I then go over to the tech tree menu and choose a specialty and get started with researching new technologies. Next I'd start planning out what solar systems you want to expand out to.

Where you go from there depends on your play style / specialty, etc.