Plenty of parking on the beach. Lot's of small tourist type shops along the beach area. In the actual city of Clearwater there is a Steinmart but that's about it for shopping.
Rather than bringing picnic lunches, head for Frenchy's on the beach. Great location and great food and drinks.
http://www.frenchysonline.com/rock_index.html
Make sure you get to the Frenchy's Rockaway Grill located right on the beach. There are 3 or 4 Frenchy's in Clearwater Beach but only one actually on the beach. It's just north of the main shopping area across the street from a small park. It's on the Gulf.
Here's their main home page with some directions and showing all their Clearwater locations.
http://www.frenchysonline.com/index.html