To answer the question, I can't see anything wrong with the grammar to your original sentence, but it feels clunky.
Missing out the beach's article is - in my experience - unusual in spoken and written English. I would only expect it when an item in a list is plural and indefinite (He sells apples, oranges and plums) or when it is a proper noun (The three biggest continents are Asia, Africa and North America). This is as these nouns often already don't need articles.
Saying "and get a haircut" makes the sentence seem disjointed and adds ambiguity (it is unclear whether the haircut is happening at the gym, the beach or somewhere else). This can be helped by instead saying "and to get a haircut" which implies strongly that this is taking place somewhere else.
I think "Next week, I will go to the gym, the beach, and to get a haircut." flows better and would be the most common similar phrasing in spoken English (at least where I live in the UK).
Alternatively, you could change get a haircut to the hairdressers/barbers such that all the items in your list are locations (as suggested by Kate Bunting). I would still suggest using "the beach".