After stalking and camouflage and so forth, your final performance depends on your pellet.
There are three commonly available types of air rifle ammunition- these are round head, flat heat and pointed. There are also hollow point and solid ammunition. Solid rounds are not common in the UK, so I will not include them here.
Round heads are the "standard" pellet type. They can be obtained in any calibre, and they are effective in all calibres. (With .177 calibre, the lighter of these pellets have a tendency to "drill" the quarry (pass through with minimal distortion, and very little hydraulic shock, probably not causing death unless a vital organ is hit). They are good for plinking, but should only be used for hunting in a .22 or larger calibre rifle. (.177 roundheads are acceptable for targeting small creatures, but should not be used on larger game unless you can shoot well enough to hit a vital organ every time).
Flat head pellets are also known as wadcutters, as they leave neat circular holes in paper targets. They are excellent hunting ammunition- fast, accurate and very hard hitting. The flat head causes high deformation on impact, passing all the pellets' kinetic energy into brutalising the kill zone- these pellets kill quickly, if not always cleanly.
Pointed pellets are also called hunting pellets, but I don't think they should be used for this purpose. Firstly, the shape of the head makes accurate machining difficult, so this type of pellet is not particularly accurate. Even in .22 calibre, these pellets tend to drill the target, only causing death if a vital organ is hit:- shoot a rabbit in the chest with a pointed pellet, and miss all internal organs- the pellet passes straight through and the rabbit lives (in pain). Shoot it in the same place with a flat head, and the shock of the impact (hydraulic shock) will cause death by heart failure.
These are broadly similar to flat heads, but they have a "lip" added around the edge (forming, in effect, a flat head with a hole drilled into the head). This causes maximum distortion on impact, as the lip flattens then the flat head itself hits home. (I have retrieved these pellets from trees and found the head flattened to around twice the size it was before firing!). Not for nothing are these pellets referred to as pest controllers. They have quite flat trajectories in all calibres, especially .177 of course, and I find them to be accurate and very efficient at killing. (Before firing, however, be sure to check the lip at the head of the pellet- if not symmetrical and "true", then accuracy will almost certainly suffer).