Your curiosity will help with diabetes. I'm am not type 2 so I haven't walked your exact path.
Testing is your lifeline to learning about how you work. You need to learn about your post-meal blood glucose curve and how various foods lead to certain results. You can only learn this if you test, keep a written record (at least for a while), and change diet as needed. There's a wise expression that you'll see written here often, "Eat to your meter." If your meter doesn't like it, cut it out and try something else. You'll discover that carbohydrates primarily drive BGs, but fat and protein play a role as well.
There are several T2s here that sport A1c's in the 5% range. Most of them limit carbs to some extent. There are lots of low-carb articles here and elsewhere on the net. It's a blood glucose, weight loss gem!
Yes, I would push for all the test strips you can get. You may have a fight on your hands as insurance companies and Medicare are getting stingy with strips. It's short-sighted policy that they will live to regret.
Test, record, learn, repeat. Don't worry about the poking, after the first ten thousand or so, it's no big deal ;-). That's actually one of the easier things to get used to.
I noticed in your profile that you were just diagnosed last month! There's a lot to know about diabetes and it will benefit you greatly to learn a few details. There's books, videos, and forums like this. I realize that diabetes is never on anyone's chosen path, but how you respond will define part of your character. You have the potential of restoring some of you insulin sensitivity together with the possibility of reducing or eliminating medications. That usually involves losing weight if needed while changing the way you eat. Best of luck to you!