Most current computers have a 64 bit capable processor and it is recommended to use the 64 bit version. The current version of Ubuntu which is used as base for this remix, has built-in multiarch support, so you should be able to install most 32 bit software if you really need it. The 32 bit version is only needed for older computers where the processor does only support 32 bit.
After having downloaded an image, you can create a bootable USB drive from it (needs a drive with a minimum capacity of 1 GB). You can do this using the USB Startup Disk Creator that comes pre-installed with Ubuntu. If you're using another operating system this task can be accomplished using UNetbootin.
Alternatively, you can burn the image onto a DVD. It is recommended to use a rewritable disc in order you don't need to waste a disc with each new version. You'll probably need the disc only once anyway. If it isn't possible for you to install from either a USB drive or a DVD, you may install a text-based system using the Ubuntu alternate installation disc and install Ubuntu GNOME Shell Remix on top of that. See the PPA at Launchpad for more information.
You should read the release notes before installing this software package.
Older unsupported versions are available, as well.