C.O.
T.:
I would seriously consider all your options. Talk to an Air Force recruiter. DO NOT sign ANYTHING until you have had a chance to think about it and get EVERYTHING IN WRITING.
If you would like to inbox me - I have a GF who was 19 when she enlisted...5 years later, she did the "boot strap" program, got her degree and became an officer, she's not a Lt. Col in the USAF - and she is one who used to work for my husband!! I can talk to her (she's currently deployed to Afghanistan) and see if I can connect you two together so she can give you her first-hand experience of the USAF - from both an enlisted and officer side.
As a single mother - you should be able to live on base - it all depends upon where you get stationed....if you are stationed at a "remote" location such as Iceland and the field you are in...you MAY NOT be able to bring your child. The military USUALLY makes allowances for these situations.
During basic training (Army and Marines are Boot Camp) you can give your mother TEMPORARY custody or GUARDIANSHIP of your child. You DO NOT have to give her full custody or sign over your parental rights. After basic training - you can figure out what is going on after you get your orders.
Serving our country is a very noble thing to do. Do it with your eyes open and get yourself informed/educated. do the research. Please feel free to inbox me so I can get you connected to my GF who is AD right now (and yes, she's married now and has 5 kids).
You can get an education and good base for future roles in your life. Will it be easy? No, not all the time. Will it be fun? No, not all the time. Will it be rewarding? HECK YEAH! If you stick it out for the full 20 years - I know a long to think in the future - however - you need to think about it - you can have a regular paycheck coming in the door and start on another career path! :)