If speed is your primary concern, call Southwest Airlines [/snark]. Just kidding. However, you need a different mindset in these days of "one-a-day" long distance trains. Amtrak trains are often late; sometimes late enough to threaten connections. Amtrak will honor guaranteed connections to the extent possible, whether this means putting you on a bus to "cut the corner" between, say, Galesburg and South Bend if it looks as if you'll miss a connection in Chicago, or else putting you up in a hotel to catch the next day's train. Please note, however, that the next day's train may not have any available sleeper accommodations, so you could be downgraded to coach. You'll get a refund credit for the difference, but you may well prefer your private room and your lay-flat bed. So an overnight layover in Chicago (on your own dime) might be the best way to nip such a situation in the bud.
With that said, based upon your given parameters the obvious itinerary is to depart Los Angeles aboard the
Southwest Chief. It departs at 6 pm and you do receive dinner upon departure as a sleeper passenger. The second day is spent traversing some pretty spectacular scenery in New Mexico and southern Colorado. Scheduled arrival into Chicago is on the third day at 2:50 pm.
Your next train would be the
Lake Shore Limited departing Chicago at 9:30 p.m. You will not be served dinner upon departure, so have a good dinner in Chicago. As a sleeper passenger you may spend your layover in the Metropolitan Lounge in Chicago (and also, as you are waiting to depart Los Angeles and/or after arrival in Boston as a sleeper passenger you may use the Metropolitan Lounge and/or Club Acela in those two cities, respectively) which you will find is much more comfortable than the general waiting area. There are complimentary snacks and soft beverages, and in Chicago you may use their showers. A small shower is available aboard the train for sleeper passengers, but a full-size shower which is not moving is a pleasant luxury. The next day you will have a bit of a boxed breakfast (a step down from the hot breakfast which is still available aboard the
SW Chief, at least as of this writing) and a boxed lunch and dinner served enroute. Your arrival into Boston is scheduled for 8:01 pm (but be prepared for the possibility of a late arrival).
As far as the fare, I'm attaching a sample Amsnag printout as of this date for October 2018 showing the various days' fare from LAX to BOS for one passenger in either a bedroom on the
SWC and a roomette on the
LSL, or else a roomette on both trains. You can see the price difference a bedroom makes! As a solo traveler there is not much reason to prefer a Bedroom over a Roomette other than the tiny private toilet & shower in your own room.
Many other permutations are possible, such as the
Texas Eagle between L.A. and Chicago or the
Capitol Limited from Chicago to Washington with a
Regional or
Acela from there to Boston, but if your aim as a first-time Amtrak traveler is to get from Point A to Point B in the most direct and hassle-free manner possible, then this is the way to go. (And, especially for a first-time traveler, I agree that this is a worthy aim.) Have a great trip!
amsnag.laxbos.201810.pdf