I have just begun to get hooked on DSLRs over the past yr. My wife had an XSi that I learned on before purchasing the T4i in June. I am looking to spend between $800 -$1200 on a new lens. I currently have the EF-S 55-250 IS, EF-S 18-55 IS and a 40mm STM. I mostly take pictures at sporting events, choir concerts (indoors with lower lighting), family events (holidays, weddings, graduations, etc.) and city landscapes.
70-200 f4 is a strong contender, though you'll want to consider the f2.8. Downside is that it likely won't be quite fast enough for indoor sports; for that I'd get the 100 f2 or 85 f1.8 (assuming you're on a budget).
I've an f4 lens (Sigma 100-300 f4) and it's good but an f2.8 aperture is missed for it's ability to blur distracting backgrounds a little more than the f4 lens and provide very high shutter speeds for sport etc.
I don't find the size of the 100-300 f4 (comparable with the 70-200 f2.
a problem, especially since the 55-250 can be brought along instead. The 70-200 f4 is a good lens but it's still fairly sizeable & weighty.
I looked at the suggestions to get a 135 f2 for indoor sports. Problem with that is that the 70-200 nails it for all but one of your uses; the 135 nails just indoor sports and means you're compromising with everything else by having the f4 lens; background blur moving from f5.6 to f2.8 is significant, the intermediate step to f4 is not so impressive.