hedwards: It takes practice. The problem with writing is that, unlike most other activities, it's more than just learning to use the hand, you also have to learn how to write. Writing lefty versus righty isn't the same and the same hand positions aren't going to work unless you also decide to switch from right to left righting.
I can write with both hands with reasonable penmanship, but I lack the patience to work on it frequently enough to make it efficient.
Other than that, you have to start with gross movements. Things like throwing and catching, those are the ones you develop first with the dominant hand, so they should be the ones you start with on the non-dominant hand. It takes time to develop an intuitive sense that you can use the other hand for the task without having to think about it.
I'm kind of a weird aberration when it comes to "handedness". I'm left-handed, but there are a lot of things I do like right-handed people. Nobody ever forced me to, that's just how it works best for me.
Things I do left-handed:
Write
Play badminton
Play softball (catch with right hand, throw with left)
Strength (my left hand/arm is stronger than my right)
Masturbate (well, why deny it)
Some tools (saws and hammers, left only. Wrenches and screwdrivers I can use with either hand)
Things I do right-handed:
Mouse+keyboard
Knife and fork
Play the guitar
I don't understand how right-handed people can play right-handed guitar. Or other left-handed people play left-handed guitar. When learning to play the guitar, by far the hardest part, and the one requiring the most strength, coordination and dexterity in the fingers, is working the fretboard. I naturally want to use the one of my hands with the most strength, coordination and dexterity for that, which is naturally the left one, since I am left-handed. Why most people do it the other way around is beyond me. I'm not alone in this either. I've met right-handed people who played the guitar left-handed for the exact same reason.