In my opinion using the standard Core.css styles is the best approach.
That way if you override the core css in SharePoint by using a theme or
a custom  CSS file the web  part will be styled appropriately as well. If
you use your own style  you would have to either distribute an external
CSS file or embed the CSS in the web part. Either one is problematic
and could be difficult to override with a custom CSS file.

The next question  is how I do use
the core.css in my code and make sure the controls  use the styles from
the core.css?

Just fill in the CSSclass attribute of the controls  you instantiate in
the CreateChildControls method of the web  part with appropriate class
names from the Core.css file. Since the webpart  is always used on a
webpart page, and those pages use a master that loads the core.css,
everything just works automatically. The trick is normally finding
which CSSclass you want to use in Core.css. It's a big file.

Heather Solomon has put together a nice reference for the WSS3/MOSS07
core styles, at

www.heathersolomon.com/.../sp07cssreference.htm

Doesn't have a lot on web  parts specifically, but it might be a good
starting point for understanding how core.css is organized.