My experience with what you're trying to do is this. I wound up going to a junior college near where I lived (Alvin Community College in Alvin TX) then I went with Southern Refrigerated Transport (Out of Texarkana,AR) to get my training done and then drove solo with them for a while.
If you don't mind let me digress here as to what the steps are.
1) get your CDL - Many options are avilable here
2) get your training with a training company, ie swift, stevens, werner, crete, etc
3) drive for a year to get that coveted experience and to payoff the training in #1.
Options for #1.
1) Go to a community college pay for it yourself and then get reimbursed by the company
2) Find a company that does sponsored training ie Swift, Stevens or CRST. A very fast paced 3 weeks and you then go work with them.
3) WIA funding for the training.
Maybe I missed it but once you get your cdl and your experience what do you want to do? Come back to expediting or stay with "those other guys"?
Oh one last word of advice, when you ask about carriers, be prepared for as many different answers as there are people. Just make sure that the people you are listening to actually worked or are working for them. Personally, I will only give my opinion on only SRT, Werner, Swift and Panther.
Good luck.