You probably won’t want to hear this answer, but nature photography of small and fast-moving wild animals is a difficult field and basically requires really expensive lenses. 500mm and 600mm lenses are commonly used by bird photographers - your typical 100-300mm zoom lens is just not long enough for great photos. And lenses longer than 300mm are both incredibly expensive and really heavy.

So the harsh reality is while you may be able to get nice snapshots of fairly tame birds with a 100-300mm lens you won’t be able to get those amazing wildlife book or calendar shots - small birds filling the entire frame - with one. You can always use your lens at 300mm and then crop off the edges of the picture, but then picture quality will suffer.

If you really want to do this type of photography on a tight budget you might want to consider buying used manual focus gear. You can find really quite decent used high-end manual focus telephotos quite inexpensively compared to their autofocus counterparts.