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Great smile! Looks and sounds like you’re approaching things in a healthy way.
The six-string barred f at the first fret is called an E-shaped barre. Some other thoughts on this chord:
It’s easier to play an E-shaped G at the third fret and even easier to play an E-shaped A at the fifth fret. One of the reasons is that the strings don’t have to make as sharp a bend at the nut (the little piece of bone, plastic or other synthetic material that the strings cross where the neck meets the peghead), hence it does not require as much strength to barre. (You can make it even easier by using a capo one fret closer to the headstock than the position you’re trying to barre, pulling the strings closer to the frets and requiring less strength to make your barre.) A second reason is that the frets are closer together.
Fretting minor E-shaped chord is slightly easier than fretting a major E-shaped chord because you don’t need to use the middle finger to fret the third string (G string). Other E-shaped chord you might want to experiment with that are slightly simpler than the major chord is the dominant 7th. Again, this is because ot requires you to fret only two strings beyond the barre versus 3. And the minor 7 requires only one additional finger to fret the A-string two frets up from the barre.
You might want to enjoy some quicker successes with these ideas and gradually work your way back toward the nut to play them as F chords after you’ve built strength playing As, Abs, Gs, Gbs etc. because you might not tire as quickly, this may actually be a faster route to playing the F chord.
But regardless how you approach it, I’m happy to see you’re doing it with good spirits.