Take the red line to the final stop south, xindian. Exit the station on the left hand side, cross to the river, and take a right, there is a bike rental place if you walk 3 minutes along the river, back towards the city. You can also ask where to go at the information desk in the MRT station.
To rent a bike you will need a passport or id card as a deposit. It costs about 60NT ($2) an hour. Helmets cost a little extra. You have to return the bikes by a certain time in the evening. I would definitely recommend asking for a bike lock. That means you can leave your bikes unattended and explore places on foot on your trip.
Along the river
From Xindian the bike route heads towards the city go towards the city, and it was a fun ride, past trees, under bridges, through puddles. There was also a little nature reserve, which we couldn't stop at because we didn't have bike locks. There are also many nice views of the river and the city too.
Treasure hill artist village
To get here it's better to cross to the other side of the river (you'll find a bridge for cycles) and then cross back when you see the artist village on the other side. This route is much more scenic and you avoid roads.