If I was injured on the job site without proper training, can I sue the company?

In most cases, no, you could not successfully sue: even if it would have been better to have trained you properly, you are under a legal duty to take care as well and not, for example, operate machinery or do any task which you did not feel competent to do. Your choice to go ahead and do the work without training could make you negligent, and your negligence could substantially reduce or eliminate your ability to recover compensation. There are times when an employee would have no way to know he was doing something unsafe, and in those cases, a failure to train could be so negligent, or careless, on the employer's part that the employer would be liable; therefore, it would be worthwhile for you to consult with a personal injury or emploment law attorney about the situation, to see what recourse you have. You need to be prepared, however, for the strong possibility that you would not have a viable lawsuit.

That said, if injured on the job, you may be entitled to Worker's Compensation and/or short-term disability, which is another reason to consult with an attorney (particularly an employment lawyer), to see if you are eligible for those.