We did this last year and it was an absolutely fantastic day and worth every penny. Like Vicky said, apart from the bit where the dolphin pulls you along, you are stood in the water. When it is your turn to be pulled by your dolphin, one of the trainers takes you to where you start, this is from a rocky ledge on which you can stand. The trainer then makes sure you are holding on to your dolphins fins before she gives him the command to go. He takes you back to the rest of your group where the other trainer will help you "let go", you are again in waist high water. My youngest DD was only 7 and she could stand in the water without having to be on tiptoe. A lady in our group couldn't swim very well and they made sure that her dolphin swim was in the shallower water. You need have no worries, the trainers look after you really well all the time. Just mention that you are not a very confident swimmer and they will go out of their way to make it easier for you. You have to wear either a life vest or a wet suit, I suggest going for a wet suit as it will have more bouyancy plus the water has to be kept quite cool for the dolphins so it will be warmer for you!
The rest of the water in the lazy river and snorkelling pool is much warmer, you can always wear a float on top of your life jacket or wet suit to make swimming easier for you.
The lunch was absolutely delicious and the whole place so relaxing....all in all a wonderful day and one I would thoroughly recommend to anyone. Also, make sure you buy some souvenir photos, I know they are expensive but they make a lovely souvenir, I have the group one of my family with our dolphin, Tyler, hanging on our bathroom wall (our whole bathroom is done out in a dolphin theme!) and it is lovely to have a soak in the bath with a glass of wine and wish I was back there again!