Only certain trains take bikes and most of the time you have to send it ahead and collect it at the station, usually the day after. I recall there were a few trains accepting bikes leaving from the station near Taroko. The west side of Taiwan is much busier and you'll have less luck.
basically you have a choice of flat-ish coastal roads, or big climbs in the Central Mountains. Taroko Gorge isn't an average gorge - it's one of the best cycle routes in the world and a place that should be included in a tour, although you could sample just the lower 30km if you really don't like climbing.
You could get by on 30 quid a day, but I would allow more if you want somewhere fancy to sleep. BTW - Cycling Plus will run an article on Taiwan soon - I think tht issue will be out in mid May.
