I had the exact same trouble likeanyother, until I found the data on Google maps.
Go to an English languuage version of the Google home page (like www.google.ca) and click on maps along the top.
Enter Taipei and it will pop up. Zoom in until you see the city. The MRT lines are on there, in the right colors, so that's a great reference point.
On the top left of the screen you'll see a link called "Get Directions" this pops up two boxes labeled A and B. The only difficulty is that the address searching only works well in Chinese (not pinyin). So what you do is move your mouse back to the map. RIGHT-click (with the right mouse button) where you want to go from and select "Directions TO Here".
Now there will be a green A (start point) and green B (destination) on your map. You can grab either point with your mouse to move it around.
Final step, under the A/B address box there should be a little picture of a bus ("Also available: Public Transit"). Click on that and voila, you can even see exactly where the bus stops are.
Of course, there's the deluxe (but expensive) version too: get a smart-phone with GPS. Then google maps will already know where you are, just click the destination and it shows the transit routes, and you always have it with you. Maybe not that expensive though if you go with a locally made phone (i.e., not iPhone or blackberry).
Or the super-duper-deluxe (and super expensive) version: the latest version of the iPhone, the 3GS, has not only a GPS but also a digital compass that works with Google maps, so it actually orients the map to match the real world based on the direction you're holding the phone.