If the computer is more than 5 years, give or take, I'd either donate it or sell it on ebay (packing is difficult due to size, costs a fortune to send 15 pounds cross country, and is probably better off being packed and shipped by your nearby UPS store. UPS will charge perhaps $20 for packing, plus shipping costs. Shipping costs at the UPS store are about 3-5% higher than you can get it directly off your Paypal screen. Ebay only offers USPS and Fedex as shipping options.
However, if I had a bunch of purchased software on the computer like Photoshop, etc, donating/selling the computer would be costly in terms of getting new software. That's a problem I had when I built a new computer to replace (augment, actually) my XP computer. The costs of replacing/upgrading the software to Windows 7 made it impractical to ditch the XP computer. So, I still have it and uses it regularly for a couple of programs that either the upgrade is very pricey or they cannot be made to operate on 64-bit Win 7. One of my friends had a JV Mcgee Bible study program that was originally written to run on Windows 3 and could not be tricked to run on his 64-bit Win-7 laptop like I did to make it run on his XP computer about 7-8 years ago. So, I 'swapped' Win-7 versions with my own 32-bit laptop, and managed to trick the antique software to run on Windows 7, although printing has to be accomplished rather oddly.
If you 'watch' Walmart and other 'store bought computer' retailers, you'll find that every couple of months, what was a $500 desktop computer is now on sale for $200-250. The manufacturers come out with a new model or two every 4-6 months so the retailers drop the price to 'closeout' sales. I found what's usually a $500 laptop for $185 about 2 years ago for a friend. Knowing the on-sale laptop was at my local Walmart, I hustled over there and and to convince the clerk that according to their website, they have 3 in stock. His screen at the work station didn't show it. But he went to another computer 75-80' away that could access the internet, went to the website, saw the computer, got the Walmart stock number, went into the back room and found it there after 4-5 minutes of searching.
Plan B: would be to remove everything from the computer (motherboard with processor, etc), disk drive(s), CD/DVD burner, and whatever else you want to carry. Pack it all tightly in a box which could be not much bigger than the mother board and perhaps 4-5" high. Remember to build crush protection around the CPU and its fan on top of it. Put the now-empty computer case at the curb and someone will likely pick it up. Then, all you'll need at your new home would be a new computer case ($50-150, generally). I'd probably take the power supply, too. But...
If you have a 'store bought' computer such as a Hewlett-Packard or Dell, you'll have to buy an identical box somewhere. The mass-produced computer companies don't use a 'generic' case, power supply, and motherboard, and may even have built-in side-of-case fans that are a fraction of an inch above the CPU fan to keep it cool. I just added RAM to a friends' 6 month old DELL computer a couple weeks ago and there's a plastic 'tube' protruding from the inside of the removable panel that comes close to the CPU fan to get that hot air directly out of the case rather than its' wondering around in the box, heating up other things, etc.
As a former package handler for Fedex, know that there's no such thing as 'fragile'. Everything gets unloaded/handled/sorted/loaded the same way. IE, all small packages go on one conveyor, 'regulars' on another, and overweight/large/hazmats go on another conveyor. I learned there to make sure everything I ship can withstand a 20' drop to a concrete floor, and can be at the bottom of a 'wall' of packages in a trailer with a bunch of 50+ pound packages on top. When the truck hits a bump, that 'jolt' to the trailer effectively multiplies the weight of the packages 300-500%, so those five 50 pounders on top of your package suddenly 'weighs' 750-1000 pounds!
Lastly, if you do carry the tower on Amtrak, not only custom-make the box for it, but create a 'handle' using shipping tape by putting 3-4 strips of tape across the topmost narrow edge with an extra 4-5 inches of each strip of tape to create an 'arch'. Be sure to put 5+ inches of tape down the sides of your 'handle'. Or, buy a cheap collapsible hand truck like those sold in an airport for $50 but at Walmart for $20 and tape the cart to the box. It'll make a world of difference if you don't have to carry it, but only wheel it.