Asus's own version of Linux is rubbish - it lacks important features like a compiler, for example. You're much better off replacing it with a modern, full-featured Linux distribution. Trouble is, the Eee series has weird hardware. There are a couple of variations on the theme of Ubuntu which have all the right drivers built-in though, so you don't have to spend weeks dicking about with kernels and configuration.
I chose to use Easy Peasy.
The Eee doesn't have an optical drive, and I don't have a USB DVD reader that I could use with it, so I had to install from a USB memory stick. Unfortunately the instructions here - which are basically "use unetbootin" don't work. unetbootin won't run on Asus's Linux, nor is there a Mac version, so it's only any good if you have a Windows machine (I don't) or another Linux machine (again, I don't). Nor do these instructions which you're supposed to follow if unetbootin doesn't work. Dunno exactly how they don't work, they just don't produce a bootable memory stick.
Eventually I gave up, took the damned thing to work, and borrowed an external DVD drive there. And then, having a Proper Linux on the Eee, I could finally produce a bootable memory stick! It is here.
To use it, dd it to a raw device (eg to /dev/sda, not to /dev/sda1), sync, and pop it into the Eee. Turn the Eee on, and within the first few seconds press Esc. Then choose the USB device from the menu and boot. You will need a USB stick that it at least 1GB, and this process will erase all existing data from it.
If you're using a Mac, then to write to the raw device, you must first unmount any filesystems on it, not by dragging them to the bin, but by starting Disk Utility, selecting the filesystem, and clicking the "unmount" button at the top. Do not click on eject. To find out which device you need to write to, use the mount command in a Terminal first.
If the memory stick still doesn't work, try putting in a different slot on the Eee, and removing any SD card you might have. If it still doesn't work, give up and use a DVD.