You could use Perl to read GPIO input (button presses) and drive VLC via it's telnet interface. The Perl modules you want are Device::BCM2835 and VideoLan::Client. To install these things from the command prompt do:
Install VLC:
sudo apt-get install vlc
Install C library:
wget http://www.airspayce.com/mikem/bcm2835/bcm2835-1.26.tar.gz
tar zxvf bcm2835-1.xx.tar.gz
cd bcm2835-1.xx
./configure
make
sudo make check
sudo make install
Install perl modules and dependancies:
sudo cpan Device::BCM2835 VideoLan::Client
Both Perl modules provide example code, the former for reading values from GPIO, The latter for driving VLC. You need to enable the telnet interface for VLC which is documented here.
If you've never dealt building an interface to physical computing I suggest a tutorial like RPi Low-level peripherals (which provides example Perl code).
The basic flow of your program is something like:
if button1 is pressed
Play video 1
if button2 is pressed
Play video 2
if button3 is pressed
Play video 3
else
Play intoductory video
To achieve this you're going to have to learn some programming skills. http://learn.perl.org/ is a good place to start if you want to use the Perl modules I suggested for reading input and driving VLC. Note that the modules have existing code, you could base a solution upon.