We are looking at investing in a memory foam (king-sized) mattress...it is the only thing that will fit up our stairs (I've heard that they come all squished up so that they fit up stairs, is this true?), and I have heard that they are comfortable. Anyone have any experience with this kind of mattress or know where I'd go to get a quality one that doesn't break the bank?

We just fit a pillow top high quality king sized bed through a small opening in order to get it into our room. The mattress is squishable for a temporary move, and the box spring comes in two parts.

A cheaper solution if you dont have lots of cash is a mattress topper. We got ours at Costco for about $150 they do have cheaper and more expensive options. Also handy if you need an extra bed you can just throw it on the floor.  
AS for a bed best bet is look for sales. I got mine at the place on rosser. No pressure from the salesman, really knowledgeable and let us try all the beds we wanted. My best peice of advice is make sure they have a good return policy at least a couple of months. You simply cant recreate sleep in a store. This will be the thing you spend most of your time in so make sure you love it. As for the boxes. I have no affiliation or inside knowledge about the link i am giving . https://casper.com/mattresses  
However it does show you how they are delivered and the size of the box. It also gives you a price point and return policy to compare locally.if you haggle a lot of the stores have beat any price by % so check all your fliers. Hope this helps.