Now that Rails 3.0 beta has been released, should I spend time building my first Rails application on 2.3 or does it make more sense to use the beta?
I'd suggest sticking with the non-beta version (2.3). Chances are tutorials and/or webcasts you might use are for the older version. Go for version 3 only if it includes features you actually need that are not in 2.3.
If you plan on actually building something with Rails, go with version 2.3 because it's, well, not a beta (stable), and because most tutorials are geared towards the 2.x series.
If you're just going to mess with it, I say go with version 3 for two reasons: you'll be better prepared when it does come out, and you get to play with the new features. But if you're not going to do anything serious with it, it doesn't really matter which you pick, so it's your call.
Your learning. Use something that is stable. Rail 2.3.x is stable and 3.0 is not.
I disagree with other answers. Rails 3 does a number of things differently than 2.3. You should learn right away the new way to code. It's not harder and you won't have time to re-learn in a few short weeks.
Rails 3 beta is stable enough. You will run into few problems.
If you are strapped on a deadline for your app, use Rails 2.3.
But I encourage you to use Rails 3.0 as it'll teach you a lot about the framework works when you do encounter problems.