OP, in the Swift system you are OTR to start. You can apply for dedicated or local after beginning to get him more frequently. However, Moose is right, you should plan on being out 4-5 weeks at a time to make good money.
I started training with Swift in early May last year, and went solo the balance of 2013. I was earning at about a $37K clip with their lower pay scale. Now it would be better. I was out an average of 5 weeks at a time, taking 2-3 days home time.
Moose did well, getting a dedicated account after 3 months solo, bringing home about $1k a week now. I landed a dedicated team account after 6 months, bringing home an average of about $800 a week. I changed to become a mentor a couple of months ago and the pay has increased to the $45-48K annualized range.