One thing I forgot to mention is that, though cutting in thirds is the way I have seen it done in person and the way I do it, you will find that a lot of people here do things very differently. And that's OK.
There are a lot of traditions in card reading in general, and Tarot in particular. Some people like to study and follow Golden Dawn teachings, some people get very deeply into A. E. Waite's teachings, or or Etteilla's, or Crowley's Book of Thoth. Others choose to follow non-written traditions, and others just find their way intuitively, or talk to the spirits, or whatnot.
All of them shuffle and cut and do everything in very different ways, and you will find here all of those ways --and more. Be welcome to pick and choose and make up your own melange.
ETA: I usually cut with the hand that's closer to the deck, whichever it may be.