If you are using Windows, just download the installer and run it; that should be very easy.
If you are running Linux, it is likely that Python is already included in your distribution. It is also easy to build it yourself with the source from the Python website. But be sure to install the Tkinter module which is needed for the GUI, too; look at the in the README file for instructions how to do that.
Python is also available for the Apple Macintosh (and numerous other platforms, actually), but unfortunately I cannot give you any further information about the installation here. You should find everything you need on the website mentioned above, though.
The uliGo subdirectory should contain the following files:
uligo.py, board.py, clock.py, unixinst.py
uligo.doc, gpl.txt, install.txt, readme.txt
sgf/ easy.sgf, hard.sgf
gifs/ logo.gif, board.gif, solved1.gif, solved2.gif, wrong.gif, empty.gif
Sgflib/ sgflib.py, typelib.py, README.txt, lgpl.txt, __init__.py
After the installation, just start uliGo, load a problem database (File-Open), and press 'Next Problem'! Most things should be self-explanatory. More information can be found in the file uligo.doc; or look at the Help menu ('Documentation').
You can use the .dat files from uliGo 0.1, though (these files contain the information about right/wrong answers etc.; for each SGF file that you used with uliGo there is a corresponding .dat file). Just copy the .dat files from the sgf subdirectory of uligo01 to the sgf subdirectory of uligo02. (In case you installed uliGo system-wide under Unix, it is slightly more complicated; please see below.)
Put the uliGo files in /usr/local/share/uligo02 (if you put them somewhere else, you have to adapt the unixinst.py script accordingly).
Carefully read, and -if necessary- edit the script unixinst.py . (I think that you probably will not want to change much.) Basically, the unixinst.py script writes a 'global' uligo.def file (in the uligo02 directory) which tells uligo to look for individual .def files (in $HOME/.uligo ) when it is started. So for every user who uses uligo, a subdirectory called .uligo will be created in the user's home directory. In this directory, the individual .def file (which stores the path and name of the SGF file used last), the .opt file (which stored the saved options), and the .dat files (which store the number of correct/wrong answers for each problem in the corresponding SGF file) are stored. In order to avoid name conflicts between .dat files for .sgf files in different directories, the path is shadowed in the .uligo directory: for a .sgf file in /usr/local/share/uligo/sgf, for example, the corresponding .dat file is in $HOME/.uligo/usr/local/share/uligo/sgf.
Furthermore the unixinst.py script creates a link in /usr/local/bin, pointing to uligo.py.
After you edited the unixinst.py script, execute it with 'python unixinst.py'. The only other thing you might have to do (if your python interpreter is not in /usr/bin), is to change the very first line of the file uligo.py, which must contain the location of the python interpreter, so that uligo can be started by 'uligo.py'.