I started this hobby in may and back then I bought my 300D used for about $900 or so. At the time I too was divided between the 350D, the 300D and going all the way to the 20D.
I decided on the 300D and only last week bought a used 20D. Im very happy with my decision, and I think if you are starting out, the 300D is no slouch. My 300D is installed with the hacked firmware, so it has almost everything the 10D has except the better focusing modes.
I brought my camera with me on an overseas work trip, and I have to say it really did not matter that the 300D was slower and older. I did mostly landscapes and some people photography. Landscapes are not going to run from you, and most people who pose for portraits will gladly wait for you to take their picture. I only felt the 300Ds limitations with the 4 shot buffer and 2.5fps when I came home and started taking lots of pictures of my daughter. Children run around and are very unpredictable.
Even then, one could argue that you do not need a machine gun to get good shots. I have to agree that with some time, you can get very well timed shots, but at other times, you just want to capture a few shots in quick succession to later pick the shot with the best nuances of expression..
From my own meandering experience, I would say go with a 300D. I got it and then later bought the 20D. It was a worthwhile upgrade to me. If you buy the 350D, you may tire of the plastic body, yet when you do, you will look back and realize that you paid $1.6k for the 350D and the 20D has a better feel but doesnt quite offer enough to justify an upgrade. You will be looking at the 5D or 1dm2 instead, which is like $5k+?
The 300D will leave you with a little spare change to pick up an excellent lens used, like the 70-200 f4L. Shoot some portraits with that lens and get a feel for what you can expect further down the road.
Whatever you decide, welcome to the horribly expensive world of DSLR photography! Lol.