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It's not a matter of which is best, it a matter of which is supported.
As of version 4.1 (I believe), the introduction of the new (2nd one) style evnironment variables supercedes the old style (1st one).
At present $HTTP_POST/GET_VARS['whatever'] are deprecated but still available (for backwards compatibility), this may not (most likely definitely not) be available in future releases.
Code being developed today should not use the old style vars as your software won't be future proof.
But you'll probably find your code easier to maintain if you use the caps version especially since you'll then have the superglobal functionality as well.
Whether you use $_POST or $_REQUEST will depend - if you want to be sure that the data has been posted from a form use $_POST, if the data could come from either the URL, a posted form or a cookie use $_REQUEST. Being explicit as to where you are getting the information from can make it easier, once again, to maintain your code.