Today we went out to the main street near our hotel and caught a cab, and I told the driver the name of the place we wanted to go. He thought for a moment and then shook his hand no no no. I said the name again, and he refused again, so we gave up, puzzled. Immediately behind him a second taxi had stopped, so I walked over to that one and said the name of the same place to the driver. He thought for a moment, said yes, and we got in.
When we got to the place, just where it was on my map, I started to give him money but then he saw something at the entry booth that I didn't catch (probably a sign I couldn't understand?) and motioned us to wait a moment while he went up and talked to the person there. He determined the thing we wanted to see had been moved to a different location, so we got back in and he took us to the new location - which turned out to be much further way.
It was a good trip, to Jeju Stone Park as it turned out, but cabs are kinda different here.
Edit: We also had a weird experience with the bus the next morning. To get to Manjanggul lava tube, Google suggested taking the 720 bus from Jeju to a place in the vicinity of Manjanggul where we could change for another bus that would take us there. We went to the 720 stop and got there a couple of minutes after the schedule said one should be there, so we thought we'd just missed it and sat down. While we were talking a bus came by, and just as we got up to look at it it pulled away and we saw it was the 720. Oh, many bus stop shelters here are built with the wall facing the road, so from where we were sitting down we didn't have a good view of the approaching bus.
But the weird thing was the next bus, the 720 we thought we'd have to wait for that we did have to wait for after all. I got on and put 1,000won in, which was what we'd been charged for every bus ride so far. As I started to walk past, the driver said something aggressive-sounding in Korean. He didn't speak any English at all and kept repeating the same thing, so at first I thought it was a higher fare and tried to offer him more money, but he wouldn't take it. Then I thought maybe he was asking where we were going, so I said the first thing that came to mind, which was "manjanggul". As soon as I said that he shook his hand in a "no" gesture" and made the money thing return 1,000won to me and pointed me off the bus.
We gave up on taking the 720 and caught a taxi, even though it as a long way.
I wonder if I'd remembered off the top of my head the name of the stop where we were going to change buses, and said that, maybe he'd have let us board his bus?