Should I use a soft cooler or a hard cooler for a beach day?

Yes, get a soft-sided cooler with a decent shoulder strap. Hard coolers are great for multi-day camping but are unnecessarily heavy and awkward to carry through sand. A good soft cooler keeps drinks cold for a full day, weighs about 3 pounds empty, and frees your hands for chairs and umbrellas. Buy one with a waterproof zipper and thick insulation — a $50 RTIC works fine.