Thanks, all. A second question then, regarding actually breaking open the piggy bank. Is there an unofficial threshold minimum price below which you get a sort of pretend tri-suitable bike? For instance, I had a 'bargain' mountain bike which weighed as much as a mountain and wasn't very comfortable when leaving the horizontal for even a slight incline. (And I don't just mean because it had me on it.) I don't want to spend tons of cash - for all the usual reasons - but I don't want to make any false economies either.
Also, on another subject, does anyone know the name of the sort of spanner that has the hole for the nut at the end of it rather than on the side? So you can shove it down a hole in the pedal to get the thing off, where you can't get an ordinary spanner because the cylindrical hole isn't roomy enough. And is there an idiots' guide to finding your way around a bike and the various basic tools it might require?