I'm not entirely clear on whether you want to survey anomalies, or just explore space.  You start off with a survey ship, a colony ship, and a scout.  You should use your survey and scout ships to explore, with the survey ship picking up any anomalies you come across (unless your ship is too weak to beat pirates).  If you want to explore faster you can design a cargo hull with an engine, a couple of life support modules, and a lot of sensors, and then buy it in your shipyard.  You usually start out with enough credits to buy a ship or two at the beginning.  I usually don't build another survey ship.

I find the tutorial less helpful than just doing a quick start.