The one with the dog running is underexposed by about a stop. Snow in photos is supposed to be white -- like the snow in the first photo. Not grey. Your first photo is perfect exposure. The dog running photo is underexposed by a stop.
If you tried to increase the contrast with post-processing, it would only make the black fur completely black out (detail is being lost in those areas already). Adding contrast is going to make the whites whiter, but it's also going to make the darks
darker -- not what we want in this case, because the photo is already underexposed. Actually, you'd want to try decreasing the contrast. (Though I realize the OP is shooting film, not digital.) I just reduced the contrast of the photo by -25 or -50 in Camera Raw and recovered a lot of detail in the dog's fur with just one slider. You can see individual hairs now, instead of just a black patch. It's brought back detail in the shadows of the photo.
So, taking away contrast instead of adding it would be the right approach, if one wanted to tackle this particular case in post processing. But best would be starting with increasing the exposure by a full stop first. When shooting in the snow, it's often necessary to compensate exposure by 1 to 2 stops, because the camera's meter gets tricked into underexposing the white snow and making it grey.
Some info here:
"PROPER SHOW EXPOSURE" - Nature Photography Techniques & Tips ©1999 Mike MacDonald
That being said, you say you are new to photography and so I think your photos are great! The dog staring right into the camera is neither here nor there, really. It's an intense look, and I'd probably prefer him not looking right at the camera like that, either. But that's your vision, not mine and not anyone else who's reading, and what's important to recognize is that you have some kind of vision. Most people do not and they get snapshots. But the first and third photos you've posted demonstrate to me that you already have some kind of grasp on how to go about achieving your vision through the camera. Just take one look around at the majority of the photos posted here, even from people who have been struggling for a while, and hopefully you see what I'm trying to say. If you're rather new, you're off to a great start, so just keep shooting.