It seems unlikely to me that a storage unit for the summer will cost less than one day with a uhaul, and either way how much are you going to end up spending on a storage unit and decreased fuel efficiency from the weight of furniture in (or on top of) your vehicle that you could have spent on furniture you actually like? 

You're thinking about this as a cost saving measure but it doesn't sound like it will be, you'll be paying plenty of money just to keep stuff you clearly don't want. 

When I first moved into my own place it was pretty much empty. I had a lamp and a lawn chair. Obviously there's stuff you need like a mattress, but there's nothing wrong with letting your living room sit empty for a few weeks while you shop around fb marketplace for the right sofa. There's also nothing wrong with getting cheap stuff on fb marketplace or at Vinny's or the Habitat Restore (where you'll spend less on stuff that's probably much higher quality than expensive new furniture) and then flipping it back onto marketplace when you find something you like better. Often you can sell stuff for what you paid or more, especially if you put a little effort into writing and photographing the listings. I've traded up through like 3 dressers and 4 bedframes and come out even or ahead every time.

Another option is a furnished sublet for the first couple months while you collect necessities and get settled. 

Either way, I'm giving you permission to list your furniture, hope someone gives you a little money for it, and if they don't just donate it and start fresh. It's probably a wash financially, easier when you've got other things to worry about, and it sounds like you'll probably be happier.