Edited - DON'T ignore the people telling you not to do it. Read all the feedback you got here, talk to reps of various military branches, talk to any friends with military experience, do your own research, & then, and only then, make your own, informed decision.
I can understand why you want to do it, but is it really beneficial for your family in the long run?
You say the dad is not in the picture. There is no guarantee (especially now) that you won't get deployed. How is your son going to feel when you get deployed & are gone, too? He needs a stable parent at home. I hear over & over again from military families that the mom is essentially a single mom when the husband is deployed, so your son would be parentless if you got deployed, pretty much. It's hard enough on kids when dad is gone for a year at a time, but I can't imagine how difficult it would be when both parents are gone.
What about reliable childcare? Do you have someone you trust THAT MUCH to watch your son while you are gone? You could be gone for months or years at a time. Are you prepared to have someone else raise your son, potentially? And would you feel fine with putting that burden on someone else indefinitely? It will grossly affect someone else's life, to drop everything to be your son's caretaker for an indefinite amount of time. Are you prepared to miss out on all those milestones, class awards, sports, achievements, etc? You can't get any of that back.
What about going to community college & working part time in your city? If you're a single mom with no job & no support coming in from your son's father, you are in a good position to get government aid, if you don't already. Most likely you would qualify for free schooling, daycare, etc. I would look into that before taking the drastic move of enrolling in the military. It would be more beneficial to your son to have at least one parent present & active in his life & not tied to the crazy lifestyle that the military can bring. I know the lure of money & benefits is tempting, but please think it through.