By the early 19th century, sea bathing was becoming super popular, enough to warrant its own little etiquette quirks. In general, there was one main rule, that men and women were separated, even if they were sharing the same beach/part of the ocean. After that rule, everything else depended on the individual bather's situation. The inexperienced bather, who was also usually quite modest, would probably opt to use a bathing machine, like so:

https://upload.wikimedia.org/wikipedia/commons/7/7b/MermaidsAtBrighton.jpg

The bather would enter and change into a bathing costume, like the woman in the right of the picture. The costume depends on the the bather's preferences. Again, the newer bathers and/or the super modest would usually wear a shift made from a heavier, warmer material, like flannel. Darker colors were also preferred. This was supposed to be warmer, although the ocean is generally cold all year round (not to mention, bathing was only "healthy" if done in the winter, when one was less likely to catch a chill), so it doesn't really make a difference once the fabric is wet. But anyway, these bathing machines were hitched to horses and then backed out into the ocean. Once there, the bather could get out and bathe/swim without assistance, like the women at the bottom of the picture, or they could have help from a "dipper" like the woman on the left. These dippers were of the same gender as the bather and gave whatever help was necessary. When one was done, it was back into the machine to return to the beach, where one could change into normal clothing and proceed with one's day.

Of course, there were many different types of bathers. Some of the more confident would often just strip (sometimes completely) and dive right in. They might even bathe in the summertime (gasp!). Since bathing in sea water was seen as a cure, some didn't even bother with bathing at all, and just drank a glass of sea water instead. I hope that answers your question and if anything's unclear, feel free to ask.