Correction: I should have said "Hualien" instead of southwest Taiwan.

Click on my name for my East Asia TR that includes Taiwan. She should buy a copy of the Lonely Planet guide if possible. Train to Taroko Gorge (the town to base in is Hualien) is very doable. So is the fast train to the southwest - lovely temples in Tainan. Doing both might be a bit much. (The fast trains go down the west coast, not east.) Not sure about the motorbike thingy - the road north of Hualien was washed out by a hurricane last year, the road to the south should be OK. There are several good day trips from Taipei.

Thanks, Thursday! I read your TR. She's not going to the southwest...my mistake...just the east and north.

Trains from Taipei to Hualien take 2 to 3 hours and run about hourly.
Tickets cost about 440 TWD ($15) each way. You can check times and
reserve tickets on the TRA website:

http://163.29.3.96/TWRail_EN/index.aspx

For some transactions, you need to know the station codes: Hualien is
51, Taipei is 100.

There's a tourist information place on the right, across from the
station in Hualien, where you should be able to find out what you need
to know locally.

You can rent scooters in Hualien, but I don't know where; you can't
rent them inside Taroko Gorge (I asked). You'll probably need an
international license. (I've not rented scooters, but I have rented a
car in Taiwan and you definitely need an international license for
those). It's about 30 minutes from the station to the beginning of
Taroko gorge, then another 30 minutes to get to the heart of the
gorge.

If you don't have time to go to Taroko Gorge, then Yehliu (also
Anglicized as Yeliu and Yelihu) is a good scenic destination near
Taipei (just Google for images). Bus 1815 from Zhonggxiai-Fuxin takes
about an hour to get there.

Thank you so much, someotherguy, for such specific info!!! This will definitely help my daughter and her friends. I think I read somewhere else that the station nearest Taroko Gorge is called Sincheng (adjust at will for spelling!). Do you know anything about that? I'm sure she could ask when she gets there. Thanks again for the help.

There is a station nearer to Taroko than Hualien but the train I was
on did not stop there and, from what I remember, there weren't any
obvious services around it (it looked more of a local "halt"). I
think you'd have to arrange something in advance (or be able to speak
Mandarin) to use that stop. Hualien is set up for tourism to Taroko:
most of the trains from Taipei terminate there and the entire
passenger load empties into the tourist infrastructure.

There are buses from Hualien into the Gorge, but I found them a bit unreliable. A car and driver would be better, especially if there's a group. I arranged one through my hotel, but I imagine any of the taxis at Hualien station would be happy to quote a price. There's a tourist office just inside the gorge, otherwise there aren't really any "services" aside from a lodge part way in, and a pricey hotel and some dubious looking cafes where the buses turn round.

Thank you both for the helpful info. Someotherguy...I couldn't find that webpage for the train at all so your link was perfect! And, Thursday...I will tell her about grabbing a driver. She's getting good at negotiating now that she's been all over! Plus...it might be better than waiting around for the bus...she's on such a tight schedule. I just heard back from her and they are adament about hitting the gorge and then renting scooters to spend time along the southeast coast. Something they read about highways 9 and 11. They think they can get gear and camp but I'm telling them to find lodging along the way. What do you think? Should there be various hotels/hostels dotting the countryside?

Well, Hwy 11 runs down the East Coast to Taitung, and Hwy 9 does the
same, but a bit inland. The distances are quite large (over 100
miles) so I don't see how you do this with a scooter. I've taken the
train from Hualien down the East Coast, where it does a bit of coast
and a bit of inland, then crosses the island south of Taitung and
heads up to Kaohsiung and Tainan on the West Coast, from where you can
switch to the High Speed Rail (HSR) back to Taipei. It's about 6
hours to Tainan, then about 2 hours on the HSR to Taipei (there's a
new connector train between the regular and HSR stations in Tainan).
Here's the web site for the HSR:
http://www.thsrc.com.tw/en/ticket/tic_how_online.asp

The East Coast scenery is OK, but not spectacular. The best scenery,
in my limited experience, is in the interior mountains, such as the
awesome road between Sun Moon Lake and Alishan.

Bear in mind that outside of the biggest tourist sites, no one speaks
a word of English.