I am starting a job that I will do from home soon.?

I will be an independent medical coding contractor. I will be doing this part time only. I have a full time job at a hospital doing the same thing. My new employer sent me a W-9 form for me to return to them. I think they will not be taking taxes,FICA, medicare out on me. Is this legal? If so how do I need to handle this to where at the end of the year I am not left paying a hefty tax bill. Do I just need to take out a certain amount and set in a savings so I can pay taxes at the end of year? I plan on going to see my accountant who does my yearly taxes when I file this year and ask him the questions but I just wanted to get a heads up on what I may be dealing with. I will be making double my hourly wage I make now per hour so working just part time I will be making about the same amount of income I work full time thus doubling my yearly income. Is there anyway I can just have my full time employer take out extra? Thanks for any input.