Pisa to Levanto by public transport on 21 August 2014

You pretty much have to change trains to get anywhere. There are 5 direct trains every day between Pisa and Levanto, early morning, late morning, early afternoon, late afternoon, late evening. If you want to get there quickly, live with the fact that you may to change trains in La Spezia, or hang around Pisa Centrale until 5:44 pm.

Trains are the best option. I doubt there are buses direct from Pisa to Levanto, maybe to Spezia, but not as frequent as trains.
The shuttle bus from the airport to the Pisa station takes 5 minutes. It's only about a mile.

kw,

It's possible that you'd be able to connect with the 13:42 departure. In the same situation, I'd try to get to Pisa Centrale as quickly as possible and then buy tickets for Levanto on the next train that's heading in that direction. That may be the 13:42 departure or it may be the 14:51 departure.

One important point to note is that the trains you'll likely be using on that route will be Regionale, so you MUST validate your tickets prior to boarding the train or risk a hefty fine, which will be collected on the spot!

Thanks for all the replies. How do I go about validating my ticket to avoid fine?

There will be a date-stamp box near the entrance to the tracks. They are often mounted on a wall and come in a variety of colors, with the older devices typically being square and yellow-orange but the newer models more modern-looking, like the photo mid-way down the page at http://www.seat61.com/Italy-trains.htm#.U-AcT1Ln_cs. Look for the phrase "Convalida Biglietti" or the newer machines also have English.