Do sound characteristics matter when moving from midfi to hifi?

Upgrading components one at a time is a good philosophy, but you won't recognize transformative sound all at once; one component will not fix all ills. Just make sure that you don't limit future purchases with an inferior purchase, save the money if you can't yet afford what you want......don't be impatient. That would have saved me a few thousands of dollars along the way.

Go to a great audio show would be my first recommendation; Axpona, Capital Audiofest, Rocky Mountain, Pacific Audiofest, Southwest Audio Show. You'll get to hear hundreds of pieces of gear, get to talk to some great folks, and get a lot of questions answered....most everyone (not all) want to be helpful

If not, then find some dealers with products you are looking for.

Personal experience is to begin with a 'practical' Integrated as the 1st course; a receiver if you must.  Try to use it to listen to as many types of speakers you accept as better than your current....  
Consider used as means of stretching the buck$....save the rest for the second course, when you can play with room acoustics and room eq tricks... ;)

Good luck and Have Fun @ It. 

If they don't matter, then why bother?

Great discussion! Everyone's audio journey is different---I started getting into hifi in the 80s (and still have the equipment). But I've also built 2 other systems along the way. One is my reference system and one is an 'experimental' system. Some pieces I bought just on reviews alone--others pieces I auditioned. Thankfully, a new hifi store recently opened up in town and the owner is very generous about letting me take pieces home to audition. I've bought a few things from him so far. Somethings that I thought were going to blow me away (based on reviews online) did nothing for me. I'm even more surprised when something lets me down sometimes vs blowing me away. Good luck and keep us posted.

But I've also built 2 other systems along the way. One is my reference system and one is an 'experimental' system.

@bluorion I also have this kind of 2-system setup now, and it's extremely helpful for evaluating components.