Which phones or PDA phones provide the best internet browsing experience?

You're looking for an app and not a phone, per se. Checking stock prices, etc. is a function provided by an app - and most apps are available on iPhone, Droid, etc.

As far as i know most of the current smart phones offer the best internet browsing and also come with lots of new features and functionalities that best fit for accessing internet.
What you desire does not come with any phone's in-built common features but your willing leads to a specific requirement and you can accomplish requirements by creating custom application for your phone. Maybe, it will be available in mobile application market.