I would say, for starters, to not be afraid to put yourself out there. I almost didn't give a demo reel to ILM because I assumed it was pointless. Needless to say, I was shocked and thrilled to find out that they wanted to hire me. Don't decide that you aren't ready for certain companies until you have at least tried. You might be surprised.
Also, there are a lot of factors that go into whether you are hired that have nothing to do with whether you have enough talent or skill to do the work. Sometimes it is simply a matter of timing, so if you don't get the job of your dreams on your first go around, don't think that you won't have other chances. Shoot for the stars!
Another thing that can be a fine line to walk with companies is trying to sound enthusiastic without sounding desperate. I've heard stories of people trying to play it cool and instead just came off as disinterested. That won't do much for office morale. Companies like to feel that you want to be there and that you want to be working on the project. It's ok to geek out a little. However, it is not a good idea to tell them that you would clean the toilets to get a foot in the door. You want them to leave the conversation respecting you a bit more than they did before, and toilet scrubbing just doesn't do the trick.
As far as progressing in your work, the one thing that has always served me well is that I always like to take on whatever scares me the most. You will learn far more by trying something and not quite getting it right than you will from animating things that are easy. If it scares you, then there is something to be gained from it. You may not nail it the first time, but you probably will the second or third time. No one in the industry asks about what your grades were or how successfully you completed your homework. They want to know if you have the skills needed to animate well, and the fastest way to get there is to push yourself beyond what you are comfortable with. Don't worry about the demo reel all the time. Try to take on shots that you will learn from, and you will probably end up with a great reel anyway.
Lastly, BE NICE! Everyone I work with is super friendly and great to be around. There aren't any jerks, because no one wants to hire them. You want your co-workers to like you, because that is the number one way that you will get jobs in the future. Personality means so much in this business, so just be friendly and enthusiastic. This job is a ton of fun, so take the time to enjoy it!!!