Start with the car in a cool shaded area.
Wash with two 5 gallon buckets, one for soap, one for rinse water. Use grit guards in both buckets to keep the mitt from ever touching the bottom of the bucket.
After spraying the car down to rinse off the dirt, use the mitt on one section at a time. After each section, rinse the mitt in the rinse bucket and then re-soap it in the soap bucket. Continue this method working from the top down, cleanest areas first.
After the wash, dry with an electric leaf blower or a high quality drying towel.
If the water spots are still there, then you most likely will need to do some polishing to remove them. Clay does not remove scratches, only things that are on the surface of the paint like sap or tar. To check and see if clay is needed, put your hand inside of a plastic sandwich bag and lightly run your fingertips over the paint. If it feels rough, then you should probably clay the car.
A word of caution on microfiber. They are not all the same. A good towel can have poor edging with polyester thread or may be hot melted which basically becomes a plastic edge that may scratch the paint. Inspect the seams and test the towel fibers by rubbing the towel dry on an old CD or DVD. If it scratches the disc, I wouldn't use it on the paint.
The above is the condensed, quick and dirty instructions, but it should get you started in the right direction for a swirl free car.
Good luck.
I've had some experience correcting washing / drying, and other defects.....
1966 Chevelle
1988 Porsche
1968 Road Runner
Randy