I shoot HFT and also vermin species on my permission - some general info for you...
.20 (5mm) is a very good pellet size for hunting if you are using an FAC powered rifle.
For sub-12ftIb rifles for hunting, then .177 (4.5mm) is the one to go for as it has a flatter trajectory and still has enough energy at 35-40 yds to take out a rabbit as long as you can make between 15 - 20mm groupings at 35 yds to make a clean kill.
.22 (5.5mm) is a good pellet for closer quarry such as rats that you would normally bait and wait for. Hard hitting but for longer shots it has a more loopy trajectory so you would need to be quite a good shot for anything over say 20-25 yds to ensure a clean kill.
You get a far better range of pellets for a .177 too.
You get a good range in .22 as well, but they are also more expensive and of course, heavier.
Try AirArms Diablo Fields or JSB Exacts for a very good quality and accurate pellets.
I have shot all the calibre's barring .25 in non FAC rifles. I would generally agree with all said above.
Just a couple of slight differences of opinion.
.177, best for target shooting, easier to shoot at longer ranges, penetrates feathers better than the others. More susceptible to wind though and at close range will over penetrate and not knock the target flat. I have shot rabbits at 15 yard clean head shots, only for the pellet to pass straight through and the rabbit run 100 yards before it knew it was dead. Not good if it gets to its burrow.Less shots per charge on pcps and a harsher shot on springers due to more force needed and decreased efficiency.
.20 IMO the most versatile of all. Choose light weight pellets and you have similar to .177 with a bit more knock down power. Go heavy weight and you have a .22 comparison.
.22. Exellent knock down ability, less wind shift, harder to shoot at long distance to start with but if your range estimation is good and you have used your combo a lot on the range then it is not really an issue. My calibre of choice. Pellets are less fiddly too which is a huge plus with cold hands..Takes down fur better. Easiest springer to shoot, more shots in pcp
All in all there is no better calibre despite what you may hear. Only different peoples opinions of the same calibres. Its all preference. All will do the job they just require different approaches. IMO .177 is better for beginners requiring less skill in the shot and range estimation. .22 will flatten anything you hit, if you can hit it. Though these days there are scopes with aiming marks on the reticule for different ranges with a .22, called MAP (multi aim point) scopes