Do I need to buy a full, brand new version of Aperture?
Yes, you need to buy a full version of Aperture 3 from the AppStore. Fortunately the new full version is much less expensive, than the upgrade from Aperture 2 to Aperture 3 used to be.
Does your computer run MacOS X10.7.4 or later and do you have an Intel computer? Otherwise you will not be able to buy Aperture 3, see the"technical specifications"
If you want to upgrade to Aperture 3, see the "Release Notes" and carefully read the instructions at the bottom of the page: "A note on upgrading". The upgrade will upgrade the Aperture library as well, and you will no longer be able to open it in Aperture 2. It is very important to backup your Aperture library before you open it for the first time in Aperture 3, and also to fix any problems with the library by running a "Rebuild Library" while still on Aperture 2. To rebuild your library hold down the ⌥⌘-key combination, while double click the Aperture Library or the Aperture Application icon, to launch into "First Aid" mode.
How would I import my existing photos from my older version of Aperture,
Just open the Library in Aperture 3 by double clicking it, and Aperture will upgrade the library to the new format. But see the paragraph before - rebuild it in Aperture 2 before doing this.
And, am I going to have conflicts by having two different Aperture versions installed in my computer?
They should be able to live side by side, but rename Aperture 2 to a name different from simply "Aperture", otherwise you will not be able to download the new version from the App Store. Also rename the Aperture library from the default name to a different name. Sometimes Aperture will create a new, empty library with a default name and might accidentally delete your older library.