For unions, though not a great advocate of them myself, you should start with Boilermakers, Pipefitters, Ironworkers, or Sheetmetal Workers.  There may be others but those will be the main ones depending upon what type of welding you had in mind. 

As far as companies, that will demand a lot of phone calls and leg work.  You will probably have to find some that have some kind of program within your geographic location and then get hired to get into their program.  That can take on many different venues. 

Hopefully there are some on our forum from that area that can be of more help.

Have a Great Day,  Brent

Boilermakers pay best for first year apprentice, well, if we leave out the 798 out in Tulsa. Generally the pipefitters, ironworkers, sheetmetal as Brent mentioned pay less. A first year apprentice with the pipefitters is around $11/hour where the boilermakers were talking a tick over $17/hour. They offer a higher percentage than the others. It's based off journeyman pay. Benefits with any Union are going to be grand.

As far as actually working for a union, I can't help you there. I've only applied and gotten accepted then turned them down as I was having more fun controlling my destiny as a self employed person rather than being told what to do.