Disclaimer
Learning to do things with the opposite hand can be dangerous. Be careful, and proceed at your own risk; I cannot take responsibility for any injuries or harm you may incur! Here are some things to watch out for.
  • Colleagues: Your friends, coworkers, or students might get annoyed at you when your handwriting is temporarily illegible or you seem to have trouble eating at the dinner table. If that's the case, take a break and around them use your dominant hand. Practice on your own for a while, and avoid making people mad.
  • Depression: One study I read suggested that training to become ambidextrous puts an individual at a higher risk of depression, and from experience I think this might be true [Sattler]. Be careful, and take a break if things start to go funny on you.
  • Sharp Objects: Knives, razors, saws, etc- be careful and move slowly.
  • Writing: Practice on your own first before taking notes in class, writing notes on blackboards / whiteboards, doing assignments, or writing notes to others. It's more important to be quick and legible than for you to get the extra practice.