In Cura, can I make my top and bottom layer be all perimiters?
I found the answer myself just after posting - I'm posting it because it might be helpful to other Cura novices.

There is a setting for this, it's just that it's not shown by default. In print settings, you have to click on the three lines next to the search box, and select "Show All Settings". Then you can find a setting called "Top/Bottom Pattern". Setting this to "concentric" does what I described. 

Actually this setting affects not just the top and bottom layers, but all layers that are part of the top and bottom shell. This seems like a good thing, but if you really want to affect just the bottom layer, there's a setting "Bottom Pattern Initial Layer" that does this. There is also a setting under "Experimental" called "Top Surface Skin Pattern" that I think does the same for just the top layer.

In addition to "Concentric" there is also a "Zig Zag" option that's quite similar to the default "Lines" mode.

You can also change the visibility of settings in the preferences menu, to make these settings show up by default.