If you check you baggage straight thru, and both your originating and destination station offer checked baggage service (which both must offer in order for them to check your bags), your baggage will be transfered between trains for you!
One note for you Bobalski: Although Boston is marked as offering checked baggage service, there are only 2 trains a day that offer it! One is the Lake Shore Limited to Chicago, and the other is the overnight Regional to New York and Washington! If you check your bags in Boston, they will be carried on one of these trains, and will get to you in New Orleans. However, it may be a day later than you arrive!
My suggestion is to hand carry your baggage between BOS and NYP - and check it in NYP!
Amtrak trains have a luggage rack above your seat (which is
MUCH larger than you will find on a plane) and also luggage storage at the end of the car! If necessary, use a Red Cap to carry your bags for you. They will carry your bags right to and from the Car, and in NYP right to the baggage check-in area. I tip them $1-$2 US per bag for their help. If your bags are heavy or large, it is well worth the money spent!
One more thing: If you are either arriving or departing via a sleeper (bedroom or roomette) into or out of Chicago, Boston or New York, you are entitled to use the Metropolitan Lounge in Chicago, or the Club Acelas (CA) in Boston and New York. There are no sleepers out of Boston (going south), but if you have a same day sleeper out of NYP, you can use the CA in BOS too!
These are Amtrak's version of an airline's First Class lounge, but do not even try to compare the two. But it is a respite from the hustle and bustle of the regular waiting area!
Finally, enjoy your trip!