You make a "reservation" for breakfast, lunch, or dinner after you board the train. It isn't really a true reservation; the dining car could not possibly hold everyone on the train at once, so there have to be multiple calls for each meal. The Lounge car is also available for snacks and beverages; some people (particularly coach passengers, whose meals are not included with their tickets) eat some meals in the lounge.
Pegg, since you are coming from New Orleans, you will be on the Sunset Limited, which runs between Los Angeles and Orlando. This is the only possible problem you might encounter - since the eastbound train must come all the way from California, it has many chances to fall behind schedule. However, there is dwell time built into the schedule, so a Sunset two hours late in Texas might leave New Orleans nearly on time.
The train carries the sleepers, coaches, diner, and "sightseer" lounge car. It is far, far more comfortable than travel by airplane, bus, or automobile. You can get up and stroll around whenever you wish; take pictures; use your cell phone (poor reception in rural areas, though); listen to a radio, etc. Persons familair with the hassles of air travel should be very pleasantly surprised. The food is usually excellent; like any good restaurant, if there is a problem they take care of it. Amtrak also has a relatively new satisfaction guarantee to assure this.
If you have any more questions regarding Amtrak or the Sunset Limited, I can probably answer them. I encourage you to try the train.