The point of going to this area is the scenery. Consequently you want to stay in the countryside, not in the city. Stay in Yangshuo. Depending on how much time you have, you can even skip Guilin altogether.
From Guilin you can hire a taxi. The ride will be 1 to 1.5 hours. From the Guilin airport it is one hour by taxi to Yangshuo. There are also buses from the airport to Yangshuo but I think that the last one is at 2h30.
Once in Yangshuo, the best transportation is to rent a bicycle and take a bamboo raft on the Yulong river.

Guilin City is definitely not the place to stay if it can be avoided. Best way to get to Yangshuo is to arrange an airport pickup from the Yangshuo-area hotel or guesthouse one is staying at. Usually this is around RMB 300-350. Guilin airport taxi mafia will not be any less.

The highway isn't too bad but the onramps seem to be quite a way from the cities. Took about 70 minutes from the outskirts of YangShuo (nearer the highway entry point) to GuiLin North station. That said, I still 2nd staying in YangShuo over GuiLin.

GuiLin does have some thing to see but the better views are nearer YangShuo.

The top photo in the OP was taken on the Lijiang (river) south of Yangdi (which is a village south of Guilin), close to where the RMB20 image was taken from.
Here is my shot at that same location:
https://stefanofoto.smugmug.com/Asia...ng%20012-L.jpg

My bamboo raft navigator held a RMB20 bill while I took the shot.
https://stefanofoto.smugmug.com/Asia...DSC_2639-L.jpg

OP’s second image looks like a classic HDR-pano similar (if not” borrowed” from) to those created by Trey Ratcliff.

Below is an example of an image from Yangshuo.
https://stefanofoto.smugmug.com/Asia...DSC_2775-L.jpg

My bold.
Excellent advice for those who has never been there. For the more experienced folks, they can stay in even more remote villages (where you can find guest houses or B&B) for better scenery. Yangshuo is too touristy for me to stay there.
Yulong river rafting is not only fun, but a photographer’s dream.

Where would I hire a taxi from? Do you think hotels in Yangshuo would be able to arrange transportation?

Also, any hotel recommendations? I'm tempted to just book the most expensive one on CTrip. I don't expect to spend a lot of time at the hotel, and I don't care for spas or anything (although a nice pool would be a plus), but I also don't want to sleep in a dump at the end of the day. I think location is very important, but I don't know enough to know if location really matters much.

You might want to check here (or Tripadvisor) as a starting point:

http://www.flyertalk.com/forum/china...-yangshuo.html

IME, none of the hotels downtown really qualify as fancy, but most aren't squalor either.

The nicer (resort type) places are outside of town, but as others have noted getting to/from town can be annoying.

Stay outside the town center of Yangshuo. Transport isn't a problem with your kind of funding--you'll be able to have the hotel arrange a driver when needed; they aren't that expensive in that area. You won't be spending that much time in the town center anyway. Try something like Yangshuo Mountain Retreat or Yangshuo Tea Cosy. Any YS guesthouse can prearrange a Guilin airport pickup for you and as I said upthread, that is the most preferable of the options.

Understood, thank you! What can I expect to pay for airport transfers and daily drivers? I'd like to have a general idea so I know when I'm getting ripped off.

I will never stay in town again during a holiday period (that was complete hell), but I can't bring myself to advise TOMFORD to dismiss the option in the absence of further research on his part.

Off hand, the pros of staying in town are:
-food options (at least some of which, are better than those at any given retreat)
-night life (not everybody wants to pass out at 10)
-better positioning for many excursions (e.g. rock climbing groups typically assemble in town)