I've been playing now for quite some time with the 21-slot ship size and would like to bolt some more modules to it. How and when do you get the next tech for upgrading?

You can unlock the next ship class with more slots and other tech from the Research section at Kite Station. As the main story progresses you'll find tech like lasers and shields etc. by exploring or by collecting debris from the enemy ships you destroy.

I already got Lasers, Shields, Super-Gun and most other technologies. Only problem, I am stil limited to 21 slots.
Can you specify when exactly in the main quest larger ships become available?

EDIT: OMG! I just discovered that you can scroll the tech-tree horizontally. Haha, finally something to invest my 7k research points in...

I was going to say that by the time I had the Stellar Class that you have unlocked, I had the other classes as well, so I thought you probably haven't scrolled to the right. :D
It's okay though. I didn't look for the Araona Capital for hours until I accidentally found it. And then I noticed the search bar in the mission log section and that that's how I was supposed to look for information for the quests...

Anyway, for everyone else wondering, at some point when you talk to Lt. Chiang and he asks for some resources in order to expand the engineering section, so it can support bigger ships.