Check out Ubuntu's documentation for details. They have a wiki and some other stuff. It looks like you've already checked out some, which is cool.
Also there a unnofficial Ubuntu guide which should be handy.
http://ubuntuguide.org
Ubuntu 6.06 is 'Dapper'. When they talk about 'support' they are probably talking about commercial support. So if you want to pay somebody to help you deal with it you can get a year-long contract. Typically you'd do this for corporate servers, business desktops, or a professional's workstation. That way if you get stuck with something you can find somebody whose living revolves around helping you and solving bugs (vs people in forums that may try to be helpfull, but can't dedicate a lot of time to you).
Also Ubuntu 6.06 they have a version called 'Dapper LTS'. LTS is for 'long term support'. This means that they will keep that version supported for the long-term... that is a couple years at least.
So if you want to have something you don't have to mess around with and do major upgrades.. to have something that will 'just work' for a long time then that is what you want. (you will still need to update you system time to time to install the latest security fixes and bug fixes. I am talking about upgrades between Ubuntu releases)
With Ubuntu 6.10 you have the more up-to-date release. This will have better hardware support and better usability (generally).
So if you want the latest stuff that Ubuntu releases you want to use Edgy. This is what most people will use for their personal desktop system, and is generally what you want to use.
So your trading the 'Long term support' for the ability to have the newest stuff, but you have to uprade in 6 months to a year when the next Ubuntu release happens.
Keeping your system up to date works the same irregardless. You use apt-get or synaptic package manager as a front end to the debian packaging system to keep your system updated and to install software. The same level of expertese is required either way.
The only 'gotcha' is that during a major upgrade between Ubuntu versions there is a higher likelyhood of you having to fix someting that went wrong during the upgrade.
For a new user you might as well go with Edgy. That's what most people will run for their desktops so you can get the best help fo that in the ubuntu forums and such if you need it. It's a lot easier for other users to help you (and you to help other users) when you have the same system the other person has.