First of all, open the timeline settings of the timeline that you want to make vertical. To do this:
1: go to your media tab in Resolve.
2: right click on the timeline that you want to make vertical to open its context menu.
3: click on Timelines -> Timeline Settings... ti open the timeline settings dialog.
You'll see that there are various settings you can change, but they are all grayed out by default because the timeline inherits from the project settings.
4: uncheck "Use Project Settings" so you can edit the settings for this timeline.
WARNING: next we'll learn about a beginner mistake I made when I started using this software!
Mistake 1: you see, when you look at this dialog, it seems you can simply change the Timeline Resolution and it will work. DO NOT DO THIS. If you change the timeline resolution, it changes the resolution of other timelines you insert into the vertical timeline, which causes all sorts of problems.
By default, the mismatched input resolution will be fixed by scaling down the other timeline to "Scale entire image to fit" in the vertical timeline. This seems to work by default and all you need to do is transform it, but it turns out that this downsamples the nested timeline. In other words, if it looks like the timeline has been scaled down by 50% by default, it's actually being downsampled to 50%, and if you scale it up by 200%, the pixels will look blurry.
Mistake 2: you would think that you can solve this by changing the setting to "Scale full frame with crop" but that actually crops the nested timeline before you can edit it, which means that the nested timeline will appear as a single vertical strip in your vertical video, and if you try to transform it or move it horizontally, you'll see vertical black bars appears as that's the black background of the vertical timeline.
The Correct Way
Instead of changing the Timeline Resolution in the Format tab, you're supposed to change the Output Resolution in the Output tab, and then use Scale full frame with crop in the Mismatched Resolution field.
Keep the Timeline Resolution in the Format tab with the same resolution as your project resolution, and then it's going to work like you would expect: the timeline will appear narrow, but you can still embed a normal timeline and transform it to change which part of the nested timeline will appear on the outputted video.
Observations
I think that I, like many people, would have expected the first setting you see to be the output resolution, so I wonder why DaVinci works the way it does!