Visit the Baja peninsula in Mexico. Tours can take you to Magdalena Bay to touch a gray whale (see Resources below for tours). The best time of year to see the whales is from January to April.
Take another trip to the Baja peninsula in Mexico. Take a different tour of Laguna San Ignacio to touch a blue whale.
Go to Tonga from August to October to swim with humpback whales. You are not allowed to reach out and touch them, but swimming with them is very rewarding. One of them may just get close enough to reach out and touch you!