While I can understand the allure of the beach, even if it's not something that's high on my agenda, I'm not sure you're going to feel it's worth the effort for just a day trip.
It's a good couple of hours there and more importantly it's a good couple of hours back again. At that time of year it's going to be BLOODY HOT, which means if you stay in the sun for more than a couple of hours you're likely to be a little pink, tired and dehydrated. Whoever drives back is going to feel the return journey. If you do REALLY want to go, I would suggest looking into booking a hotel on the beach for one night(keeping your existing accommodation), it's going to cost you about $70 I'd guess. That way you can pack just an overnight bag, enjoy your day at the beach, have the use of a hotel to shower freshen up and drive back the following day nice and clean but more importantly fully awake.
IMHO the problem with going to a public beach in Florida in the summer is if you want to escape the BLISTERING heat, it's hard to do. If you book a hotel (with a pool) you at least have the option of scurrying back to the hotel if it's simply unbearable, cooling off a little and returning to the beach/pool later when you feel a little better. At that time of year the Atlantic coast is a little cooler both air temp and sea temp, therefore may be a better choice. I would agree Cocoa is a little tacky, but I think that;s a fair comment on Clearwater as well. Nicer areas would be Captiva, Anna Maria Island and Naples on the Gulf side. On the Atlantic side I REALLY like the DVC resort at Vero Beach, Disney do a great job of making this a really top class hotel, other options could be St Augustine to the north of Florida.
I'm not a beach person and would much rather spend the day at Typhoon Lagoon splashing around in the surf there and swimming with sharks :O LOL