When I turn the wipers on in the car the blades make a skittering noise across the windshield when it rains.
First try washing the windshield with metholated spirits or some other cleaner. Be sure to get rid of all the spirits, usually a damp cloth will do the trick. Make sure you do it in the shade. Oil build up on the shield is a come problem. It also maybe time to renew the wiper blades.
Some Methods I Found; Vaseline will stop the screeching, but with some aggravating results after years of messing with aftermarket blades and refills, i just go to my honda dealer and buy the rubber inserts made for the blades. take the old inserts out, remove the 2 spring steel pieces from each, put them in the new refills, then reinstall. seems silly to spend over $10 on 2 tiny strips of rubber, but the wipers work like new for a few more years Rain-X? But then you don't need the wipers. Oh well, no more screeching either way. Even when the Rain-X coating is in too poor a condition to deflect rain adequately, it may lubricate the glass well enough to stop the noise
sounds to me like you got a used tire, am I right? if so the tire they put on was defective in the fact that the tread was separating, thereby causing the shaking feeling you had upon leaving the tire shop. if it was a new tire then you have a real complaint against the tire shop.
The rim is most likely damaged and now unbalanced. You should look into replacing the rim.
First you need to thoroughly clean your windshield and the rubber part of the wiper blade. If they still do it try replacing the blades. If it still does it you may need to replace the actual wiper arms. They are the part that connect the wiper blades to the wiper motor. They contain a spring that helps to hold tension on the blades so they dont skip across the windshield. As with any part those springs inside can wear and lose tension allowing the blades to skip. The arms are a fairly inexpensive part depending on the make and model though you may have to go to the dealer to get them.