F of all, you can get an awful lot done with G, C, and D7. After that, it's good to learn D, Em, and A. A 3-finger D is fine, no need to use the 4-finger shape that works as a movable chord shape.
The F shape is important but it's not easy. The way I tell people to practice it is to alternate repeatedly between an A chord (very different from the F shape) and the F shape. You'll have to constantly re-form the F shape and that's the essence of what you need to practice. Change chords back and forth, and settle into a steady pace, like a rhythm. However slowly you need to do it to get the F shape RIGHT, that's how fast you should do it.
To check for correctness, don't just strum the chords, but play the strings with a slow “rake”, so you can hear each string individually. If you're not getting clear tone on any string, correct your fingering until the tone is clear. Very gradually, try to increase the speed of your chord changing. Look at your left hand to see which fingers are the slowest and least accurate. Have a little talk with each problem finger, and tell it you expect it to hustle more, and be there correctly when it's time. Then watch it and see that it does just that!
Working with a metronome helps here. You could set it to 30 (a change every 2 seconds) and see if you have to move it to a slower tempo, for it to match the time your hand needs to do the job right. Once you determine the speed your fingers can do it, stay at that speed for a while, and concentrate on making it right every time. You'll probably need to rest your hand after a minute or two, as the muscles, that haven't had to work so hard before, start complaining.
After a rest, do the back and forth, A to F, some more and see if you can increase the metronome speed slightly. Glare at the problem fingers, probably the ring or pinky, and tell them to hustle. ("Get there FIRST!")
Regular work on this exercise will definitely yield results. When the F shape goes on quickly, you can change chords more quickly, with less looking needed. That's a very important skill for backup, because if you have to look at your left hand, you can't look up to see the chord changes the guitar player is doing, which you have to follow.
Keep after this problem, and it should improve noticeably in just a few weeks. Good luck, and happy jamming!