(first post... woohoo!!!)
As a working professional in the games industry and an animation instructor, the best thing I would recommend is developing a strong portfolio and/or demo reel. It's not the software you use, it's your work. It helps to know Max, Maya, or whatever a particular studio is using, but it really does boil down to your skills both in 3D and more traditional art (drawing, sketching, photography, painting, etc). If you're an animator, put together a demo reel that shows the best of the best of the best of your work. Polish your timing, posing, staging, conveyance of weight, acting, etc. If you're a modeler, show off a range of characters and objects in both hi- and low-poly states. If you're a rigger, show off some of your raw rigs and how they're used on sample characters. There's really no definitive answer to getting a job, but this is a good start.
The suggestion to make MODs is a very good one. That's a great way of learning a bit about what it takes to build game-ready shapes and animations. Another option is getting involved in the indie games community. There are a lot of independent teams trying to make their own games, some using commercial engines, some making their own. Regardless, it could prove to be a good way to gain some game development experience.