I'll start in the beginning of the next year (2010) a big new project with Rails. Can somebody give me advice which version of Rails I should use and why?
I would go with the latest version (2.3.5), since there don't seem to be any signs of an imminent production release of Rails 3.0, which was announced over a year ago. I'd also suggest trying to use Ruby 1.9.1 over 1.8.X, since the performance increases are substantial.
The latest bit of news regarding the Rails 3 release date has been ~first-quarter 2010, so it's a good time to put your feet in.
The developer-facing API is said to be stable, so if your project doesn't rely on too many plugins, or if you're willing to put in work to make plugins Rails 3 compatible, I say it's safe to go with Rails 3. And if you're not already, go with Ruby 1.9.
Now, that is what I plan to do for my next project, but I'm also happy with fixing any problems that will arise. If you're still a little unfamiliar with Ruby, or don't have the time to roll up your sleeves and fix the things that need fixin, go the safer route: Rails 2.3.5 + ruby 1.8.7.
If this is going to be a production project I would definitely go with 2.3.x and 1.8.7 - they are proven, reliable and will work with the majority of gems and plugins available.
On the other hand if the project is more experimental then it may be worth looking at Rails 3, I would not however choose it for a commercial project unless you are experienced enough to deal with the potential compatibility and other issues that may be part of working with it early on.