When you decide to become a truck driver, your family will be affected greatly by your decision. Generally, drivers with families, especially young families, would be advised against seeking over-the-road driving jobs. That said, after their first year of schooling, training, and experience, it will be easier for drivers to find a position which brings them home more often, so to some, the sacrifice is worth it.
Regardless, we have plenty of resources for drivers with families (and pets) to help them understand the lifestyle and make informed decisions.
Most new truck drivers will start out driving over-the-road, obviously with some exceptions, for at least their first year. Most large carriers will give one day of home time for every 5-7 days on the road, usually with some kind of limit.
The hardest part for any truck driver with a family, or anyone doing a job that keeps them away from home, for that matter, will be the time spent on the road away from them. Advances in electronic communication have made it easier to talk to and even video-chat with family, but it is still a major consideration to make for anyone entering the industry as a new driver.
Company policies on having riders along in the truck will vary wildly, but will normally be restricted to family members above a certain age, who don't hold a CDL. Many companies may require a fee to cover extra insurance, or may restrict riders to certain times of year, i.e. summer.
Attending a private school will usually mean either going part-time or on weekends or at least being within driving distance of home and/or current place of employment.
Most company-sponsored CDL training will not start paying until the driver has his CDL and is on the road with a company trainer, so arrangements would need to be made accordingly. That said, the trucking companies are constantly changing their policies and you might be able to find a few that pay, at least a little, during schooling.