I have heard many people describe the animation industry as 'cool', 'interesting', 'glamorous' and many such similar terms. Well, no wonder 3D animation jobs are so highly sought after these days.
Depending on which part of the world you are in, there is either a shortage of job vacancies for animators or an even bigger shortage of job vacancies for animators. In short, there are more people applying to become animators than studios offering positions.
In such a competitive environment, this is where a CG artist has to excel in packaging and promoting himself. You will have to learn the art of networking first of all. In many instances where the bigger studios are concerned, submitting showreels might not be sufficient to warrant the HR's attention, because big studios means lots of work and that means BUSY. If you participate in trade events like software launches or trade exhibitions, chances are you can get to know a lot of studio executives. By getting to know somebody directly and having a name to contact, you might get your demo reel viewed much faster as 'you know somebody'. And if your reel is any good, chances are you will get hired, especially if the studio is in a hiring mode.
Now sometimes even after you've done the above and your demo reel is really good, you still may not get a job, and that's because the studio just isn't hiring at that point in time. However, your portfolio will be kept for future references when they do have a vacancy.
Also, there are times when studios just don't want to hire fresh CG artists. They may prefer to hire 'experienced staff'. But who's going to give you experience if you don't get hired by them, you ask.
OK, to answer your question, first you must realize that there are a number of industries where 3D animation is used, and you may be able to get started in an industry other than the one you initially had your heart set on. If you can't find a job with an animation studio, then consider working with a web design company, a game design company, an architectural firm, a media company, an advertising company or even a product design company. Many of such companies require a 3D artist, and you can build up a portfolio and the much coveted 'experience' there.
Getting an internship during your education is also a good method of paving your way to getting your first animation job. If during the internship you exhibit yourself to be a conscientious worker with good habits and skills, you might just get a job offer upon graduation. Many 3D artists got their jobs this way, so this is a very valid method of getting your first job. Another spin off from this method is by offering to work for free. Look upon it as an opportunity to learn from working professionals and to gain experience with the latest softwares and real-world projects. If you show yourself to be sincere and dedicated enough you will eventually be offered a position.
Whatever the case, remember to get a demo reel produced. The reel should exhibit your best work and your skills to the fullest. Take out the crappy work and if there's nothing good to put in, then work on getting some good stuff to put in instead of putting in crap. Crap is not going to get you any job offers! A demo reel is going to make or break you, so make sure you put in 1000% effort into making the best reel you can produce. Know that the person reviewing your demo reel (if he ever picks it up) is going to spend less than 30 seconds looking for something he likes. The moment he feels bored or unimpressed he loses the incentive to continue watching your reel and you have just lost your job offer. Make your reel as interesting and as impressive as possible. Leave what happens next to providence.