How should I go about installing Ubuntu on a Cubieboard?
If you want a current version of "Ubuntu" a.k.a Vanilla Ubuntu with Unity and a Vanilla kernel, You don't setup Ubuntu on a Cubieboard - That being said, there are some Unsupported EOL Linaro based images that will get you a Lubuntu-ish desktop on them.
Fist I'm going to list out some of the resources you may want to look at:
http://cubieboard.org/ This is the main site for the cubieboards, The English is roughly translated, but this is the first place to see news about the boards, It also has downloads of all the images on http://cubieboard.org/download/ (ignore the pictures and scroll down for a list)
http://linux-sunxi.org/Category:Community_Devices wiki style general information about the boards
There are two ways to install/run ubuntu on these boards. First you can flash the firmware replacing android on the device - which is a complex process (If you really want to try this way, try this with one of these downloads most likely the 1.06v) . Or you can install on a microSD card. I recommend trying to use the SD card.
Then do these commands in terminal(replacing sdX with your microSD card, BE CAREFUL I WARNED YOU)
cd Downloads gzip -d cb-a20-lubuntu-desktop-card-v105.img.gz sudo dd if=cb-a20-lubuntu-desktop-card-v105.img of=/dev/sdX bs=1M sync
Pop the microSD card in the cubieboard and boot. (if you get "display not supported" or something like that you might need to ssh in or blindly get to tty and try switching the display, I think this page had the command: http://docs.cubieboard.org/tutorials/a20-cubieboard_lubuntu_desktop_releases)
And the rest is too much of a mess to even mention, oh and this will only get you a linaro version of "Lubuntu 12.10" don't expect any graphics drivers or acceleration or anything to work properly, and if you want flash try this: http://hwswbits.blogspot.com/2013/04/flash-on-picuntu-linux-arm.html
And I left out the cubietruck because I don't own one.
Another option is Armbian. Despite its name it allows you also to build Ubuntu-based images. No surprise there, as bootstrapping Debian and Ubuntu is a very similar procedure anyway.
At the time of this writing you will, for example, find ready to use Xenial-based (Ubuntu 16.04) images with a range of different kernels at the above linked location.
I've run my own builds in the past (the Armbian project supports a wide range of ARM-based boards) and you can effectively hook into just about every stage of the build process by means of shell scripts. So this is highly customizable if you take the time to read up on how to roll your own.
Oh, and yes, while the kernel isn't the stock Ubuntu one, the repositories used by an instance created from these Armbian (Ubuntu) images uses official Ubuntu package repositories.