Ds1 was petrified of getting water on his face, up till age 6, then 'got it' and now at age 8 is a fish - can swim well.
Dd learnt to swim at age 5, no fears (except of the swimming teacher - we had a term where she didn't want to do anything with the teacher watching), is now at age 6 extremely confident and proficient.
Ds2 is 3, confident in water as long as he has armbands, doesn't like water on face though.
Because of where we live mine have all been around pools since about 6 weeks old. One loved it from the start, one didn't and the littlest at 2.5 has started swimming on his own this week.
At any rate, imo they are not really able to swim properly until they are about 4.5/5 and it makes no odds really as to when they start before this age. So don't feel bad about not taking your toddler swimming.
I would recommend going when you feel happy about going, so that your dd doesn't pick up on any anxieties you may have. Also, don't go for the first time during any school holidays, as in my experience, public pools in school holidays are hellishly busy and noisy.
Don't go for very long, maybe 20 minutes max and make sure you have something for her to eat afterwards as they get very hungry.