This is coming from a decade-long vet, recently out of service and back in the civilian life, so take it for what it's worth.
So, congratulations on joining the military. It's a big step to take and can seriously turn your life upside down... hopefully for better (like me) and not for worse. I just hope it was the Army because otherwise we need to spend time 'measuring dicks' after you're out of training.
That being said. Training sucks. It will always suck. You may say to yourself now "Self, it will get better after I get out of AIT." My friend, that's when the real fun begins, but not the fun most people equate to 'fun'. It's the kind of fun that you tell yourself is fun when you look back on it, but for the most part really sucked when you were doing it. You see, right now you're in a controlled environment. One in which the NCOs (or whatever your leaders are called in your Service) are told "don't break the little Private's (Seaman's/Airman's) pretty little heart and will right now, there will be enough of that when they get to their 'real' unit". So... any hardship you find now, learn from it. Absorb it. Make your skin that much tougher because one day you'll be out of training and someone will need you to prove they can rely on you to save their life. And they're going to MAKE you prove that through training.
Keep in mind, I say this with a smile on my face because really... when you're done you'll feel like one accomplished mother fucker. And you'll get where there some humor behind the above paragraph (mostly because it's true) and appreciate it.
As for being away from home, coming back on leave and then leaving again... you'll get used to it. I'll be honest, it will steel your heart for deployment and such later, just like training will. It may feel weird and emotional now, but very honestly you'll learn your way around it. It's rough the first couple of times, but I'm positive you'll get through it.
Regarding your writing? Just do it. And if you can't find yourself able to finish with someone else, break it off... or explain to them what's going on and ask them to take a break with the intentions of going back to it.
Once again, congratulations and thank you for your Service. Make yourself proud in the military. I look forward to more stories once you're out of training.
-Nasty