Also, do shipping lines only transfer installations that are already constructed on my homeworld to colonies
Yes, provided you set up supply/demand points in the civilians tab.
can they build their own facilities?
Civilians will build their own mining outposts, who's mineral output you can purchase, and they'll produce their own infrastructure and ship it to colonies that need it automatically, but all other installations you have to build yourself.
Anyone have any tips for me in regards to designing warships?
It's more efficient to limit your ship classes to one role, and have that role in mind while designing it. Do you want your ship to launch antiship missiles, launch antimissile missiles, shoot lasers at enemy ships, provide point defense, acquire targets for other ships, launch fighters, or something else? Each one of those designs will require different kinds of sensors, weapons, or engines, so they'll be more efficient if you specialize them.
The other big part of ship design is just making sure that some of the numbers line up. Weapon range and tracking speed should match firecontrol range and tracking speed, sensor range should at least be equivalent to weapon range (and usually it should be several times greater), estimated deployment time should match up with estimated maintenance life, etc.
The best way to learn is to design a ship that you're happy with and then post the design here so you can get some feedback.