You can install both!
Mostly, many codes work in both too.

As often in these "what is best" questions, the answer is: "it depends"...
If you want a strong statement, I would say to start with the latest version, ie. 2.0b currently.
After all, if you are hooked, you will want to follow the latest versions, know the quirks of this line, and the new ways of using Processing.

Of course, as you found out, there are some gotchas, hence the "it depends": on your needs, on what you want to do.
You might want to use the 1.5.1 version if:
- you need to export a Java applet;
- you want the old P2D / P3D modes;
- you want to use a library not compatible with 2.0
and so on.

Personally, I have both on my computer, and I can run them side by side if needed.

Thank you for your reply, i will continue to learn with the 2.0 version.

The b in the version name stands for beta, so, yes, expect bugs and stability issues. Now, a software always has bugs, but it is said stable when there are less bugs reported, or they are minor / in corner cases...

The OpenGL side (everything except the Java2D default mode) is less stable, as they are actively working on it, to improve performance, iron out wrinkles, etc.
There are some things to get used, like manually importing Java classes that were imported automatically before: lot of people were lost when trying sketches working in a previous version of  Processing (eg. using Calendar or similar).
But overall, it is worth going on with the current version.