Generally speaking there are a couple different types of schools you can attend. Honestly this isn't necessarily that critical of a decision. I have known tons of people that have gone through one type or the other and both seem to do just fine. There are a couple things you should know though.
Private schools are not associated with any particular trucking company. They are run independently so upon graduation, students can choose to go anywhere they would like. The schools range in price generally from $2000-$5000 and can take anywhere from 3 weeks to 8 weeks to complete for your Class A license. It would take less time and money if you only wanted a Class B, but I strongly suggest you avoid going that route. You are not only embarking on a new job, it's a new career.
By only getting your Class B license you are severely limiting yourself not only with regard to the number of jobs available to you but the amount of money you can make. It only costs a little more money and takes a little bit more time to get your Class A license and believe me, you won't regret it. Almost anyone I have ever spoken with who had a Class B license had plans for getting their Class A, but I've never once talked to anyone that wished they only got their Class B. Nobody. Go for your Class A.
Many of the better private schools will have a wide range of training. The training will include several weeks of practice backing up the truck, classroom study and lectures on everything from log books to maps and routing to air brakes and vehicle inspections. They will also have several weeks of training out on the local roadways.
Make sure they will give you the necessary training to get ALL of your endorsements including hazmat, tankers, doubles and triples, and passengers (buses). Make sure you get ALL of the endorsements. Each endorsement only requires a very short written test. You can also severely limit your job opportunities if you leave them out.
Be sure to ask the school what their policy is if you don't pass the driving portion of the test for your CDL the first time. A reputable school will continue to work with you for a short period of time to allow you the extra practice you need. They can't guarantee you'll pass it, but they will continue to help you for some time.
Also, most good schools have a job placement program. There is a HUGE demand for drivers nationwide and their job placement department will have a long list of companies willing to hire you the day you graduate. You can tell them the type of job you are looking for and they will have several choices for you. Believe me, getting a job coming out of school is very, very easy. But getting the right job for you at the right company will be much more likely if you have some experienced help with good contacts.
Many schools will also offer financial aid. The money you are required to put up front is sometimes a big limiting factor on what type of school you can attend. Offering financial aid can open up your opportunities. Find out how long it will be before you are required to begin making payments after graduation. Most schools will allow you 3-6 months before you start making payments.
The advantage to attending a private school is that the training is general for the industry, not specific to one company. Therefore it will apply no matter where you choose to work a will not be geared toward any one company's particular equipment or agenda.
The disadvantage is that it may cost you more money up front if you aren't eligible for financial aid.