In real terms I dont think you will have any focus issues with the 55-250 its a nice little performerand its focussing is completely silent. Now for a little more info, which I dont think will be relevant to your shooting style unless you shoot really fast sports where servo focus speed is critical.
The STM lenses work best (and are designed specifically to take advantage of ) on cameras with contrast detection sensors as opposed to phase detection sensors. Contrast detection was first implemetented on the 650D to take advantage of the STM drive, it utilised a Hybrid AF Contrast and Phase Detection sensor. It also features on the 70D etc. Unfortunately for you the 60D was just a few months too early and still uses Phase detection only. Now dont be too disappointed though because it will still focus quickly and although dedicated sports photographers will say USM is faster on Phase Detection cameras (and they are correct) I think in average real life terms its not noticable.
So still agood and worthwhile purchase in my opinion. I dont own one but a few years back I had the original 55-250 for a while and it was astonshingly good for the price. A little gem of a lens.