I am new to receiving 1099-misc. How do I go about setting up regular payments to the IRS so I avoid getting penalized?

I am a personal trainer for a gym. The gym was recently bought by another company. The new company is treating us like independent contractors and thus not taking out any taxes, social security or Medicare. I am learning that if I continue to work here, I should be paying these items throughout the year so not to get penalized at tax filing time. My employer has not told me any of this.