The F chord is a tough one and is a lead-in to barre chords. You've hit a point where all players hit a bit of a wall, so don't worry. The good news is that once you get the F chord down, barre chords won't be as tough of a nut to crack.

Something that may help you out is realize that it's OK if your index finger goes out beyond the B and E strings and touches the G string, or the D string for that matter. This is because you are fretting those strings with your other fingers higher up the fretboard.

With that knowledge, you may find you can get better leverage with your index finger across the B and E strings.

Oh, one other thing, you need to build up some strength in your fingers/wrist/arm for these types of chords. So part of it is to practice and over time you will develop the strength. Don't worry too much in the beginning that you can't play the chord totally cleanly.

This tutorial may help:
http://www.guitartricks.com/tutorial.php?input=1624

Good luck.