This is not a fast or easy process, so you will need to be prepared to put in many hours of practice if you plan on becoming ambidextrous. Developing these motor skills will probably give you a whole new appreciation of what babies’ lives are like.
If your writing is very messy in the beginning, start by tracing large text out of a book or magazine. It may also help to buy children’s paper, which has widely spaced lines for large printing and dotted center lines to control the letter proportions. Another good thing to do is to observe the way left-handers write or just ask them for some tips.
You should also practice writing the most common words in your language and your name, as this will teach your muscles common letter combinations. Lists of the most common words in each language can be found on Wikipedia. Be prepared for the fact that your left arm and hand muscles will be quite sore after writing practice. This is because you are training certain muscles for the very first time.
Stick people, square houses with rectangular chimneys, round-headed cats with triangular ears…the goal here is to become more dexterous, not to produce a Rembrandt. Try coloring them in as well to make you feel more comfortable with your left hand. Also, try to draw straight lines from left to right using your left hand. It will teach you to push, not pull.
You can just write backward (from right to left) or you can practice mirror script, where the letters themselves have flipped around. Writing backward is also helpful because you will not smear the ink or tear the page when you write with a pen—however, it will not be that easy for others to read, so try to save it for your diary (just like Leonardo da Vinci!)
This makes writing more comfortable and leaves your hand less likely to cramp up at the end of your practice session. Be sure to use quick-drying ink though, or the text may get smudged as your left-hand moves across the page.
If you open doors with your right hand by default, start opening them with your left. If you usually take the first step on a staircase with your right foot, do it with the left. Keep working at it until leading with your left feels natural and easy.
Eating your food (especially using a spoon). Blowing your nose. Scrubbing dishes. Brushing your teeth. Dialing phone number and writing SMS on a cell phone.
Tracing is a great place to start: having a defined edge to work with will help force your eye, which is visually tracing the outline, and your left hand, which is physically tracing it, to work in sync. Trace your right hand onto a piece of paper. Pushing the pencil against 3-D contours will help guide the left hand. Graduate to tracing 2-D images. You can think of this as taking down the gutter guards at the bowling alley.
The thumb is used in almost every situation you use your dominant hand. Not being able to move it freely is an excellent way to make you aware of all the times you are using it—so try tying your right thumb to your right index finger with a piece of string. You could also try wearing a glove on your right hand or putting your right hand in your pocket or behind your back.