By Scooter
The Gorge is not the best place for a first scooter experience. However, it's a fantastic place to ride for those with experience. Scooters are available to rent in Hualien. If you run low on fuel, the locals often stock a small supply that they'll part with for a miraculously uninflated price. There is also a gas station 50km passed Taroko on the number 8 road around kilometer 118. The drive there climbs to an impressive 3 km altitude. Bring warmer clothes and rain gear. Also make sure to get a helmet with a rain visor. You can rent a scooter from Pony near the train station in Hualien around NT$400/day
By car
Renting a car in Hualien is definitely an option, though be very cautious; the roads through the gorge are extremely narrow with numerous bends. In addition, there are pedestrians, scooters, cars and massive tour buses all vying for the tight space.
On foot
If you do not have any of the above you can see some parts of the national park by foot. The closest hiking trail to the park entrance is the Shakadang Trail, which departs from the road after the first tunnel approximately 1 km from the park headquarters. The eternal spring shrine is only 1.4 km further trough the tunnels. Taking this tour takes about 4-5h, don't forget to take enough water with you.