Bradley,
I have never done "shutdowns" strictly as a welder but more as a millwright (electrical work, mechanical, fit, fab and weld). My little bit of experience with it involved a lot of thinking on your feet and being faced with problems that need solving quickly. Usually a lot of installs/repair and a lot of on the spot fabrication to make that install go in on time. There are a lot of companies that specialize in building equipment and then going to site to install. Hooking up with one of these companies might be good way for you to get into that line.
My advice to you is you got to start somewhere ....don't be afraid to go out and try (and try for)any type of work. You won't ever know what suits you and what you want to make a career out of until you get a taste of it. If you plan on answering shutdown calls out of a newspaper or trade paper...spend some time talking to the contractor and be upfront with them to make sure that they will be willing to take you on for the whole job before driving xxx miles to the jobsite. Also it is a good idea to check on the rep of that company any way you can. One last thing Bradley and this is the MOST important...A lot is happening on a typical shutdown...it can be a highly dangerous enviorment and on big ones you won't know or neccessarily be able to count on a lot of the people your working with....you are the guy thats gonna make sure you stay safe and live thru it in one piece.
Good luck
Tommy