No argument that the Amtrak site leaves much to be desired, however this is a six-segment trip with flexible dates. You'd likely need to do a significant amount of investigation to build the optimal trip no matter what mode(s) you choose. And optimization in this case doesn't just mean absolute lowest fare, but other factors like how many days at each destination (not too few days, not too many), possible concern about connecting times, willingness to ride coach, etc. also come into play. It would be an easier task if Amtrak gave a fare calendar so you could see which specific days were cheaper, but that would just make the process a notch or two less cumbersome. Your best bet is to think of this as six individual trips and stack one on top of the next.
Others have already contributed some thoughts but here are mine. Note that I mostly checked between mid January and mid February, so fares may vary of course.
Minneapolis-Philadelphia
This is around $850 for a roomette and roughly 30 hours with two connections, one in Chicago and the other most likely in Pittsburgh or Washington.
Leg 1 Minneapolis-Chicago is about 8 hours and daytime. If you want to save money you could do this leg in coach and save roughly $250 or so.
Leg 2 is from Chicago, most likely the Capitol Limited. It's a 2:45 connection but it's pretty reliable -- 93% of the time successful in the past 3 months.
You can take the Capitol Ltd either to Pittsburgh or to Washington where you can connect to a Philly train.
Leg 3, no matter if you connect via PIT or via WAS, this third leg of the trip is on a train without any sleepers, so you'll be in coach. It's a daytime segment.
--connecting via PIT, this third leg is 7.5 hours in coach and you arrive about 3pm
--connecting via WAS, this third leg is about 2 hours in coach and you arrive about 5-6pm
Note that going via PIT looks to often be nominally less expensive (well under $100) but via WAS has more time in your roomette including meals.
Philadelphia-South Carolina
Apologies if I missed your specific SC destinations, but its roughly $500 and 11-12 hours in a roomette for most destinations. If you're going to Charleston or elsewhere on the Palmetto and want to save money that offers a daytime trip for about $200 in coach. The trains with sleepers are all overnight and depending on where you're going you'll arrive in SC in the very early morning.
South Carolina to Denver
Fly. You can typically get a four-hour nonstop flight from Charleston to Denver for under $150 per person on Southwest with enough advance purchase. The rail trip is around 55-60 hours and a roomette can easily be $1300-$1400. I'm certain many on this board would love a trip such as this, even when it's sandwiched into such an long venture like this one. But even if you fly this segment you'll still have dozens and dozens of hours onboard Amtrak during your odyssey. And if you're new to Amtrak you may not want to risk growing sick of train travel so early in the trip.
Denver-Sacramento
31 hours, one train, scenic, easy, and I found some days under $700
Sacramento-Portland
15.5 hours, one train overnight, easy, $400-ish
Portland-Minneapolis
37 hours, one train, two nights. This is the one I saw the most variation in fares -- some days I found an $836 fare but some other days were several hundred higher. Because this could easily be your most expensive segment and seems to be the one with most fare variation, my suggestion would be to start with a cheap day for Portland-Minneapolis, what will be your final, homebound segment. Then use that date and work the rest of your itinerary backwards. For example February 12 has an $836 fare, so then work your way backwards and look for a cheap fare Sacramento-Portland in the range of February 7-10th, etc. Depending on how much "play" there is in the number of days in each city you can hopefully get good fares for every leg.