How to teach a five year old to swim in a pool?

My family always goes to the beach for a week on the fourth of July. My little sister Jaina, who turned 5 in April, likes it just as much as everyone else. The condos we stay at have a great pool, where Jaina can usually be found in the shallow end with her floaters on. My mother suggested I teach Jaina to swim this year, so I intend to do that. However, Jaina is afraid to go underwater or have anyone let go of her in water from her waist up. I am desperate to find a way to get her to trust me in the water. We bought her a floatie suit/vest which she is excited to wear, but I need to teach her to swim without it, without pressuring her and it not being fun.