Did people 'go to the beach' for pleasure during classical antiquity?

Yes, they did. Cumea for example, was a popular resort for the rich and famous. They fled the summer heat (and the malaria) of Rome to the beach.

I don't have any references, but some (very) rich Romans had heated seawater swimming pools constructed on their properties.

Going to the beach is not exactly what we do today. Going back a wee bit in time: my grandfather lived in the Dutch beach resort Scheveningen (period 1880-1900). In summer he rented out rooms to tourists. I've seen old photos in which people were dressed in bathing costumes (with long sleeves for both arms and legs) wearing hats went in the water. That was about it; they splashed each other with water. Some swam a bit. People went into the water in enclosed carriages. In the carriages they changed clothing and went into the water.

Sunning was frowned upon. The general believe was that a suntan was something bad and besides, only peasants had a suntan. You don't want to look like a peasant, do you?

Sunning on the beach, playing beach volleyball or other games, huge crowds on the beach in Roman times - forget about it. The Romans, most certainly the upper crust, didn't want to get suntan. They weren't racist in the modern meaning of the word, but definitely didn't want to look like sun burned peasants or slaves.

The beaches wouldn't have been crowded with holiday makers. People had to make a living. Only the wealthy could afford relaxing in a beach resort. Everybody else worked.

What did they exactly do? I have no idea. But I can make educated guesses: first of all, we're talking about the very rich. That's not a big group. Most very rich people got to be wealthy and important later in life. I can't imagine Roman senators or business tycoons racing each other to the beach. They probably strolled along the beach - which was fairly empty. Some fishermen at work likely, perhaps a few people offering snacks, refreshments, etc to them.

I'm sure their younger offspring would have had fun on the beach. But again, not exactly as we perceive beach holidays today.

West Mersea

There are several examples available of Romans taking up residence near the sea for recreational purposes. One such example comes from West Mersea, were it is thought the Romans built "holiday homes", which definitely suggests leisure to be one of their reasons for being there.

West Mersea Seaside Heritage Project page 5

Because of its close proximity to the Roman Colonia of Camulodunum (Colchester), West Mersea was extensively settled by the Romans, indeed some wealthy citizens may have had ‘holiday homes’ on the island. In the central historic area focused around the Church of St Peter and St Paul and West Mersea Hall, archaeological excavations have unearthed mosaic floors

BAIAE

Another such example is a place on the coast of Campania called Baiae.

According to the Dictionary of Greek and Roman Geography, this resort was spoken of being a place of luxury for Roman nobles.

Dictionary of Greek and Roman Geography

BAIAE (Βαΐαι: Eth. Baianus: Baja), a place on the coast of Campania, celebrated for its warm baths, as well as for the beauty and pleasantness of its situation, on the SW. side of the bay between Cape Misenum and Puteoli, which was commonly known as the Sinus Baianus. We find no mention of a town of the name in early times, but its port was celebrated from a remote period, and was supposed to have derived its name from Baius, one of the companions of Ulysses, who was buried there. (Lycophr. Alex. 694; Strab. v. p.245; Sil. Ital. 12.114; Serv. ad Aen. 6.107, 9.710.) But it was never a place of any note till it became a favourite resort of the wealthy and luxurious Roman nobles towards the end of the Republic: a favour for which it was almost equally indebted to the abundance and variety of its warm springs, and to the charms of its beautiful situation. Horace speaks of the bay of “the pleasant Baiae” as surpassed by no other in the world (Ep. 1.1, 83);