I'm not the expert since I stayed away from the engineering building as much as possible as a cadet, but I asked the Engineering Officer on my cutter and this is what he said:
All Engineers minus those who decide to go to flight school or want to be a Deck Watch Officer on a ship start out as the Engineering Officer in Training or EOIT. Your first tour last two years. After that, engineers have several different options they can pursue. Engineering majors who love the afloat life are still eligible to be Patrol Boat Executive Officers and start tracking to be an underway Engineering Officer. The EOIT that reported to my ship from my class has had a lot of responsibility over the major systems onboard. As an Ensign, he was responsible for running our 60-day major dockside period where he oversaw contractors and CG civilians and managed over $1 million in repairs...this was about 10 months into our tour.
If you decide that the underway life isn't for you, civil engineers can apply for transfers to Civil Engineering Units (or CEUs). CEUs are typically smaller units focused on facilities engineering. The Coast Guard operates 30,000 buildings and structures including roads, lighthouses, towers, runaways, etc. Eventually, you have the opportunity to apply to attend Graduate School on the Coast Guard's dime and earn an advanced degree in some type of engineering. If you continued with civil, the next career path would be Executive Officer of a CEU, facilities engineer at a major shore command, and several others.
I used the magic of Google and discovered this guide:
https://www.uscg.mil/hq/capemay/AdminServices/doc/OCareerDevGuidebook.pdf
It's a little outdated (1998!) but it provides a good base on each of the officer career specialties available to members. Page 80 will get you to the Civil Engineering section.
I've also attached a file with a few pages from a more updated Career Guide from 2012 focusing on Civil Engineering. It provides a career progression pyramid, answers some FAQs, and provides a pretty good description of what its like to be a Civil Engineer in the Coast Guard. I hope this helps!
If anyone has anymore questions on specific career specialties, I'd be happy to answer them or pull out some more references from our CG network and upload them here!