The easier approach would be to update Ubuntu (10.04 will only be supported by Canonical for a few more months).
The safest way to update Ubuntu is via Update Manager, but you will probably only be able to upgrade by one or two versions at a time.
If you want the latest Ubuntu directly (12.04 Precise or 12.10 Quantal) you can download it from the Ubuntu site then install it. This is best if you just want to start afresh and you have backed up all your files and data to an external drive. When you have 12.04 you can use the Ubuntu 2.0.0 package of Audacity and when you have 12.10 you can use the Ubuntu 2.0.1 package of Audacity. These Audacity versions are regarded as quite safe to use.
If you want an easy way to use the latest “alpha” Audacity (not a “release” version) then on 11.10, 12.04 or 12.10 you can install instead the Ubuntu PPA Daily Build of Audacity “2.0.2 + SVN” from here: https://launchpad.net/~audacity-team/+archive/daily .
At the moment the Audacity versions posted there will be regarded as very safe because they are very like the code Audacity will be releasing soon as version 2.0.3. But if you installed the “Daily” next month, it might have bugs and experimental features because the builds would be between releases. You have to decide this for yourself.
I’d recommend doing the fresh install if you can. Upgrading by multiple versions can be very slow and things can go wrong because so much stuff is being updated. Doing a fresh install is likely to be a lot quicker and easier.