More reach will serve you better than faster glass at this point, and I speak from experience. A 70-200mm lens just isn't long enough to capture wildlife most of the time. I started shooting birds with one and was constantly frustrated by the lack of reach - and adding a teleconverter only made things worse, not better.
The 70-300 IS was my second wildlife lens and was well worth the money. Be prepared to shoot at about f/8 for improved sharpness, which will be difficult in low light, but otherwise it does a good job. The IS is invaluable and it's relatively small & light, which is something you'll miss if you ever upgrade away from it.
Even at 300mm you'll wish you had more reach, but the same is true at 400mm or 500mm. But it's still enough reach to make getting the shot possible much of the time. And given the uses you mentioned wanting it for, I think it's the best place to start.