Must an employee be paid for travel and training time?

If you are an hourly wage employee, and you have to take mandatory training for your job (and it has no use for anything else in your life), the time you spend taking it is considered to be compensable work time.  Your employer has to pay you for it.  Also, if it puts you into an overtime situation, then it has to be paid as overtime pay.  Additionally, any such training being conducted in an off-site location, must also be paid by your employer. If you are not being properly paid, then you can contact your state department of labor, or contact an employment attorney for help.

If you refuse to attend this training, then yes, you could lose your job.  Unless you have an employment contract, union agreement or there is a company policy contrary to this (or if some type of discrimination is a factor), you must take any training that your employer requests.  The fact is that if you are an "at-will" employee (i.e. none of the aforementioned applies to your situation) your employer can hire or fire you for any reason or no reason, as well has increase/decrease salary/hours, promote/demote, and generally impose requirements as it sees fit.  You in turn can continue to work for your employer or not, your choice.