The list is on the righthand side when you open a shipyard.  You can scroll up & down to see all your choices.  More choices appear as you research new technologies.  Good ones to use when you don't new new fighting ships are missions - treasure hunt is always a low cost high reward pick - you can never have too many credits.

Another thing to remember, and you may already have done this, is when you research bigger and better ships is that you'll want to upgrade your fleets.  Don't forget to decommission old outdated ships as you do this, it saves you upkeep on potentially useless hulls. 

Doesn't matter which version.  It's not about upgrading ships you already built (which can't be done in any version, except for commander ships).  As you research new techs bigger, more powerful ships will become available.  As I said, send out research missions if you can to speed the process.  Another trick is to build supply ships to help your core worlds grow faster.

Just a helpful note, always reference which version of the game you're playing (base vs Supernova) so the person trying to help understands where you're coming from   

You can build expeditions or research missions, so that you still use your factories, but do not multiply your ships.