I try to take a couple long train trips every year, so I am pretty familiar with Amtrak equipment. For the Southwest Chief to Chicago, the family bedroom is on the lower level of a two-level car and will sleep two adults and two children. There is no bathroom in the family bedroom, but there are three bathrooms and a shower room on the same level as the room. The Lake Shore Limited between Chicago and New York has single level cars and no family room, only bedrooms (which include bathrooms) -- for four people, you could get the bedroom suite, which is two bedrooms that open up and are combined together. If it's just you, the spouse, and the two-year-old, a bedroom would probably be sufficient.
A less expensive option that we use sometimes is to get two roomettes across the hall from each other. Each roomette sleeps two people, and two roomettes together is often less expensive than one bedroom. You can leave the roomette doors open if you want to see each other and talk, or close them if (older) kids need a break from parents or vice versa. (We are a couple with no kids and sometimes need a break from each other!) If the kids are small, put one kid with one parent in each roomette overnight. You can book the two roomettes online, but then call the 1-800 number right afterwards and have the Amtrak customer service person modify your reservation so that the two roomettes are opposite each other.
Amtrak has been getting better with the delays, but the Southwest Chief and Lake Shore Limited both run on tracks owned by freight railroads who sometimes prioritize their own trains over Amtrak. There are also delays that can be caused by weather (flooding, tornadoes), or by idiots on the tracks who get hit, and then law enforcement has to do their thing before the train can move again. If the delay is long enough to cause you to miss your next train, Amtrak will put you up overnight and will put you on the next train, but you risk losing your sleeping car accommodations if the train is sold out, and you would then have to travel overnight in coach. Time permitting, we always overnight in Chicago so as to avoid that risk. Lots of wonderful things to see and do in downtown Chicago, so it's a pleasure, anyway.
If you think of the train trip as part of the vacation, Amtrak can be very mustachian. For sleeping car passengers, your meals are included (except on the Silver Star along the Atlantic coast), and you aren't spending money for hotels since you are sleeping on the train.
My friends on the Amtrak Unlimited Discussion Forum (
http://discuss.amtraktrains.com/) will be happy to bombard you with answers to any Amtrak questions you could possibly think up!