With the phone, you can practise composition - without a doubt the most important aspect of any good photography. You will also get a feel for the sort of features you want in a more highly specified camera when you decide to move on, based on the type of photography you want to do.
When you want to get more creative, you will probably want to be able to control shutter speed and aperture (amongst others). There are many highly specified compact cameras that allow this - but avoid models that make you use the menu system to set them - ideally you want dials/controls on the camera body for this.
However, there is one important restriction with digital compact cameras, and that is the ability to create "shallow depth of field". This is when you have the subject sharp, but foreground and background blurred, to make the subject pop out (Often used for portraiture).
Here is an example. You can see the eyes are sharp, the background blurred. There is not much foreground, but even the nose is starting to go out of focus.
This effect is not possible (or at best very difficult) with digital compact cameras. Without getting technical, this is due to the sensor size (typically 1/20th surface area of a 35mm negative, or smaller), which nearly all digital compacts have. To get this effect you will either need a DSLR, or one of the very few (only 2 or 3 that I know of and very expensive) digital compacts with larger sensors (typically slightly less than 1/2 surface area of a 35mm negative).
Some links on composition
Rules of photo composition Digital Photography Composition Tips
Photography: The Rules Of Composition (Technology: Digital Cameras: User's Guide)
See also
Digital Cameras: User's Guide - advice videos on Digital Cameras: User's Guide
