As a QA Manager in a fabrication shop, I would say concerntrating on FCAW-G and SMAW with E70XX minimum, in all positions and SAW, handheld and track, would cover most all structural/bridge applications. Broadening that to the field, you could include FCAW-S, again in all positions. GMAW may also be used in some shops, but not any come to mind around here.
Tests that are required include 1" plate, with backing, in the 3G and 4G positions to cover all positions and 1/8" to unlimited thicknesses. Of course SAW in 1G position only.
Getting certifications through AWS (which I am not personally experienced with) or an outside agency or union hall may help you get your foot in a door, but more probably you will have to qualify/certify with the employer that hires you.
Hope that helps you get started.
My vote would be to take the one inch structural plate test(s) per AWS D1.1. SMAW and FCAW for field work and FCAW, maybe GMAWS (Spray) for shop work.
Most states require the welder to work on a state bridge project inorder to maintain the certification once it has been issued. The certification will lapse if the welder isn't specifically welding on a bridge project covered by DOT.
Maintaining certification to the D1.1 Structural Welding Code/Steel is usually easier. The welder simply must show continuity by maintaining a welder's log of the projects he/she has been working on to demonstrate no periods of inactivity.
Best regards - Al