Well your first problem is that you should be using a Nikon.
The usuals apply - YMMV, I'm not a pro but I did sleep at a holiday inn express last night...
If I were you, I'd seriously consider taking a digital photography course at your local community college. That'll ensure you have the basics down first if you're experienced with film SLRs, but not digital SLRs. They'll probably also convince you to shoot in RAW if you're not already doing so.
Now on to your results. The graininess is mostly due to the fact that under lower light, if you still want to run faster shutter speeds, the camera is likely (you're shooting in auto mode?) stepping up your ISO, making the image sensor more sensitive. This is what introduces the grain. The only way I know of to combat it is to use the lowest f-stop (aperature) on your lense and/or manually control your ISO, taking the camera off auto mode. If you're zoomed in to 300mm, that is a 5.6 f-top on your lense. Even when zoomed all the way out to 75mm, it is still a relatively high 4.
So purely from an equipment perspective, look for a lense that has a lower f-stop (lets in more light) at the zoom levels you need. These get expensive pretty quickly...
2569A004 Canon EF 70-200mm f/2.8L USM AutoFocus Telephoto Zoom Lens with Case & Hood - USA