I think this should do it:
Open the terminal with Ctrl+Alt+T, or search in the dash for 'Terminal' and open it
Then enter the following:
sudo apt-get install firefox ubuntu-restricted-extras
You will be asked for your password to give it permission to install the above stuff
apt-get is the command line installer included in Ubuntu. You could also use the Software Centre if it is installed. See the manual for the Software Centre.
firefox bit installs Firefox, whilst the
ubuntu-restricted-extras bit installs packages restricted for copyright and closed-source reasons, including Windows fonts, Java, and Flash Player. These are not always included by default.
Bear in mind that you cannot get the latest versions of Flash after 11.2 on any Linux system.
If you really want the latest version of Flash, you may be able to use the Windows version, as you can for Silverlight.
For the latest version of Firefox, see Ubuntu documentation on Firefox New Version.
ubuntu-restricted-extras does not work, try using this, which should install what that package should have installed:
sudo apt-get install ttf-mscorefonts-installer unrar gstreamer0.10-plugins-bad-multiverse libavcodec-extra-53 gstreamer0.10-plugins-ugly gstreamer1.0-plugins-ugly adobe-flashplugin flashplugin-installer gstreamer0.10-plugins-bad gstreamer1.0-plugins-bad gstreamer0.10-ffmpeg gstreamer1.0-libav gstreamer0.10-fluendo-mp3 gstreamer1.0-fluendo-mp3 chromium-codecs-ffmpeg-extra
Though this should just install the flash player:
sudo apt-get install flashplugin-installer
And this might also work:
sudo apt-get install adobe-flashplugin
For any of the above, make sure all the repositories are enabled in Software & Updates: