When dealing with chord troubles, the best thing is to always make sure the guitar is set-up well. This means having the neck looked at to ensure it is in good shape, as well adjusting the action, intonation, and all the other good stuff that goes into guitar maintenance. This can be checked in a one-stop trip to a music store and than you are good to go.

Once you know the guitar is in good shape, it simply takes practice, practice, practice. The "F" is very important to learn and get down because it helps open the door for the use of "barre chords." Unlike G, C, D, Emi, etc; the notes in an F chord do not lend themselves to an open position shape and you simply can't get out of getting that barre chord shape down. Once you do, you can play an endless amount of songs and chord shapes. Keep at it.

The barre chord and the shape played on the 1,2, and 3 strings are from the same shape. The latter is simply a condensed version. Honestly, either will work but for the sake of practice take the process slow and work to get the barre chord shape down. You will get it with time. You are developing new muscle memory and you simply need to get it under your fingers just like you did when you first picked up the instrument.

Again, make sure the guitar is set up well; and than dig in to those barre chords and give it time. You will get there. Best of luck! :)