It might be stating the obvious, but you always want to pay some attention to your tires before taking your car out of storage. In addition to visually checking for any cracking, dry rotting, and that they still hold air and have sufficient tread, it's a good time to take a look at their manufacturing date codes. On cars which don't see much mileage it's not hard to lose track of how long ago they were last changed, and all of a sudden you've got tires that are past their best-by date.
Like the associated storage article suggests, I like to inflate my tires to close to their maximum sidewall values before winter. That makes it easy to confirm that they still hold air as they should in the spring, it'll make it easier to spot any bulging, prevents flat-spotting while the car sits, and it's easier to bleed off excess pressure than it is to pump the tires up when you're itching to go on that first drive. If you have a spare tire, it's a good time to check its condition and state of inflation, too.
If you have a modernish fuel injected vehicle that has a clear-flood mode, an easy way to build up fuel and oil pressure before the first start is to simply press the gas pedal to the floor while cranking. This accomplishes the same thing as unplugging the coil and/or injection relay, but is easier.
I bought new lower priced tires last summer for my 300SL and they are fine except they seem a little on the soft side. They seem to get a small flat spot after a week but they smooth out after 3-4 miles. So I put the car on blocks for the winter to make sure I did not have a real problem this spring. Come on sunshine!
I don't think it's good for the suspension to leave it completely unloaded during the winter. A good balance would be to get most of the weight off the tires, while still leaving them in contact with the ground to keep the suspension loaded.
GM bulletin #13-03-10-001A: "Information on Tire Cold Weather Cracking" – (Jan 30, 2014) advises “avoid driving, moving, or test-driving vehicles equipped with high-performance summer-only tires below 20º F as operating at these temperatures can cause damage to the tires.” The car models affected include the 2012-2014 Buick Regal GS, 2014 Cadillac CTS Vsport, 2012-2014 Chevrolet Camaro ZL1 and Z/28, 2014-2014 Chevrolet Camaro SS 1LE, and the 2014 Chevrolet Corvette Stingray and Stingray Z51, and SS sedan. My Continental Extreme Contact Sport suggest the same below 45F.
Some good advice, thanks for that, but I add one step before cranking to put oil pressure around. I disconnect the coil lead and also remove the plugs to reduce the possibility of wear occurring on "dry" crank bearings with piston/cylinder pressures imposed on them. For the few minutes required to remove the plugs on my simple toys I feel it's worth the little extra effort.
After finishing the 7-year restoration on my car (https://www.hemmings.com/stories/tag/corvair-homecoming) in September, I got a little paranoid. Just parking it in the shop would result in dust from our rural area, there are insects and mice to deal with. After all the effort and expense, I wanted No chance of any problems. Bought a car cover -- not a custom-fit -- and while cruising Amazon came across a return of an inflatable bubble at a nice markdown price. After its first car show post-restoration, all that was necessary was a little cleaning. Checked the fluids (There aren't many - it's a Corvair.) and inflated the tires properly. Filled the gas tank with non-EtOH premium and added a can of Seafoam. Drove it for 7 miles to ensure proper mixing through the fuel system. Parked it on cardboard under the tires to ensure no adhesion to the plastic floor of the bubble. Disconnected the battery to ensure the clock wouldn't run it down while it sat for several months. Put the cover on, then inflated the bubble over it. The fan has a filter to keep dust and insects out. There's about a foot of space between the bubble wall and the car, so there's no chance of an inadvertent bump against the paint while in the shop. Of the half dozen mouse traps around the car, only one activated during storage. Yesterday, I deflated the bubble and removed the cover, reconnected the battery and the car fired up immediately. There was no dust on or in the car, no sign of any insects. The paint is as shiny as when it was put inside. I haven't driven it yet -- won't while the roads are wet anywhere. But I am most pleased at how well it came through storage these past nearly 6 months. Admittedly that's a bit over the top, but this is a special car to me, and will be the last special interest car I will ever own. It's ready for the show season.
I never retire mine for the season. If it is reasonably nice and the roads are salt free, I'm riding. If I can't get them out for a run, I am at least running them up to temperature every couple of weeks. The worst thing you can do to a car is let it sit