I think that Sigma 70-210 is quite an old design so it doesn't quite live upto modern standards.
The other problem with it is that it doesn't have image stabilisation (often labelled as IS on Canon lenses, VR on Nikon, OS on Sigma and VC on Tamron lenses). Stabilisation stops minor movements, wobble and hand shake in the lens affecting the sharpness of the image.
When using a longer lens (ie more than 70mm) you need to have your shutter speed faster than a reciprocal of the lens focal length.
So, lets say you have the lens zoomed to 200mm, you need a shutter speed greater that 1/200th of a second to give you the best chance of making sure small movements of the lens don't affect the sharpness. Image stabilisation can help with this, but it's good practice to keep the shutter speed up.
The other problem with wildlife is that it moves, unless it's an animal asleep on the floor you probably need to be at 1/500th of a second to be in with a chance of freezing the animal. You'll need the shutter speed even faster if it's a deer walking or running. Best bet is to follow the deer with the lens, try to wait for it to pause for a moment, then fire the shutter.
If you were in full auto mode, the camera will have made decisions about what settings to use to get a well exposed image (ie not too bright or too dark) but it won't have known what your subject was.
You could put the ISO on auto and then put the camera on Tv on the mode dial, this is called "shutter priority" you can then set your shutter speed to be at whatever you want, say 1/500th, then the camera will pick the right ISO and aperture (f-stop) to expose the scene.
If you're photographing in the evening, it's likely to be getting dark and the camera will have slowed down your shutter speed to keep the ISO closer to 100 (for less noisy images). With nice evening light, you might still need your ISO at 800 or 1600 to get a decent shutter speed.
As for lenses themselves. I have the Tamron 70-300mm f4-5.6 VC (ie the model with stabilisation), this can be found
second hand for about £150, I prefer it to the equivalent Canon 70-300mm IS, which I've owned before.
You can also look at the Canon EFS 55-250mm IS, which is smaller and meant to be pretty decent, but might be a bit short for wildlife.
As mentioned above, you can look at the Sigma 150-600mm (there's also an older 120-500) and Tamron also have a 150-600. These will get you a lot closer, but you're looking at much bigger lenses and spending £4-600.