Welcome to the industry gatevalve1982!
I also work for a nuclear engineering company as a design engineer. I have done design engineering for the last 12 years and I love it.
One thing I have noticied for our new hires, since we are an engineering contracting company, is that it takes longer for them to get real plant experience since we work out of an office and don't always spend time at a site. I am lucky in that I spent my first decade as a design engineer at a plant, so I got the opportunity to get involved in almost everything. If you are the kind of person who likes to get in the field and learn, then I would encourage you to find if an opportunity exists for a staff aug or long term deployment position at a plant. The more you get out there and see actual plant modifications being installed, the better you will be at your job. Then you will also get an opportunity to see all the other types of jobs there are at a plant (systems/programs/reactor engineering, operations, maintenance, rad protection, QA/QC, NDE, etc..). The plants have a lot of engineers that support day to day operation of the plant, and that can be a lot of fun, if you are willing to work nights and weekends, and donate your entire life during a refueling outage.
The existing fleet of plants will be around for quite some time, so you can have a long and full career just on these. They may be older designs, but it provides a lot of opportunity for design changes due to the need for upgrades to equipment. Most of the original fleet has been or will be relicensed, so they will be around for a while. As an electrical engineer, one thing that you might find interesting is that some plants are starting to replace old analog instruments with digital equipment, which is a really new thing for our industry.
The new plants are also an exciting opportunity, especially when they actually start building them. The flood gates haven't opened yet on this. The AP1000 being built at Vogtle (in Georgia) is the only one right now that is proceeding full steam ahead. There are several others that are in the works, like the one that my company will build, but the recession has slowed things up a bit.
So my advice is to get to a site, talk to people, spend time in the plant, suit up in scrubs and PCs, get some zoomies, and learn as much as you can.