To web browse with the 420G, press the globe button on the keypad. It is the key above SEND.
If you are getting "Connection Failed", it is possible that the carrier has not yet activated Mobile Web for the phone. After a new phone is activated, TracFone's web site states to wait one hour before accessing Mobile Web.
If you continue to get "Connection Failed" after a period of time, then you should call TracFone to get it rectified.
I noticed earlier today that the TracFone default home page that appears, had some server problems and displayed an "HTTP Error 500". Going to the Navigation selection on the menu then selecting Reload, will reload the page.
To save any favorites, you would save them as Bookmarks, once you are on that particular page while web browsing. To look at the Browser menu without launching the browser itself, press the Menu key then 5 (or navigate to the center icon).
T-Mobile customer since 2005.