The increase in weapon quantity is not so gigantic.
However the tactical (red) skills you can put into your ship will allow you to briefly boost the firepower of your weapons so you can do all those things you saw others do.
you get more weapons slots as you level up. im assuming your a LT or an ensign. higher levels get more weapons slots on ships, of course depending on what type of ship you get. dont quote me on this, but every tier higher works as so- escorts have more forward weapon slots than both sci and cruiser. cruisers have same amount of fore/aft weapons slots, and sci get the $*!% end of the stick and have fewer weapons than both cruisers and escorts. again dont quote me on that because i only play escorts.
Yes, the starting ship sucks and everyone is stuck with it at first (unless you paid for the NX). But once you move up to Lt. Cmd you can pick a ship better to what your playing as or what your fighting style is like. I'm an Eng but when I reached captain I moved away from Cruisers to a Sci Intrepid class and used it all the way up to RA2. The better ship you get the more options you have with it. More slots for consoles to beef up your weapons and abilities and more weapons slots. Is 4 the max? I think so, but I've never flown a tac ship so I don't know if they have more or not. Just keep in mind what kind of weapons you want to use because you will be better off specing your abilities towards them as you move up in rank and not trying to just become good with all of them.
It's true, the starter ships aren't all that wonderful. They're meant to be simplified so you can focus on learning the basics of how to play the game, what the powers and power levels do, that sort of thing. Things open up a little more at LC level but you'll have to make CM before any of the "cooler" stuff really becomes available.
Just grind on it and be patient... we were all new to this once. If you play regularly you should be able to make Admiral in around a month.