If you want indoors, you're going to need speed.
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Just this past week I shot some random sports using either a FA50mm f/1.4 or the kit 18-55 AL II (forgot to bring my 70-300).
Even in a large sports gym with lighting from the ceiling and diffused from two rather large banks of windows, the kit lens was just too slow to shoot satisfactorily. The 50mm fared a lot better, but even then I was up to 1600 ISO the whole time, in order to freeze action (dodgeball / basketball games) for the most part. I kept the FA50mm at around f/2.4-2.8 to get max sharpness while still retaining around 1/300 shutter times.
From my limited shooting experience, you're going to want a mid-tele for "events", like watching local baseball / football games, indoor events, etc.
For larger stadium events, you might want something longer. For birding you
definitely want something longer. Many birders will say you want to start at 300mm and go from there.
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Also, remember, you get what you pay for (usually) when it comes to lenses. Take my Tamron zoom for instance -- It's cheap and goes to 300mm, but I'm picky and never use it past 220mm or so, as I cannot stand to see the drop in sharpness. Also, it's sharpest when stopped down to about F/8 or so....shooting wide open gives me a noticeable drop in sharpness.
So, while my lens is specced for 70-300 f/4-5.6, what I use it for is a 70-220mm F/8 and macro. This doesn't bother me, though, as it was only $129 U.S.
Of course, I may be more of a pixel peeper than most, so take what I say with a grain of salt
A "do it all" lens (something like a 18-250mm) is going to be convenient, but you'll sacrifice speed and quality to get that convenience.
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Personally if I had the cash, I'd have a DA* 300 for larger sporting events, I'd then add a 1.4x TC for birding, and the DA* 50-135 for smaller events or indoors. Throw in a fast ~16-45mm lens for most wideangle shots and you're set.
I plan on definitely getting the 50-135 here real soon, as I think it will fit most of my shooting style. It's pretty sharp even when wide open, so it'll probably end up being my indoor lens, portrait, and event lens.