How can I blow up the Pirate base in the tutorial?

I'm three hours into the tutorial and can't see ANY decent ships.    I've got these little ships that are pretty useless - they certainly don't scratch the paint on that pirate base. 

So what's the trick?   I've looked through the research tree for something like 'large hulls' or 'Battleships' and I can't find any way to build a decent fleet.   

I still need a few more colonies and to blow up that Pirate base and that's all I have left to do in the tutorial. 

So any tips on how I actually go to war and not get blown to bits?