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So I have finished the tutorial missions, and I was using the number pad to steer my craft, control to fire, and A to accelerate. I felt like I had a hard time moving and hitting targets. I do not have a joystick and do not plan on getting one. I have a Xbox 360 controller, but I do not feel like I would be able to map enough functions to the buttons.

Do you guys have any tips for flying that might be able to help me, or will I just get used to it over time?
If you have a mouse, I'd suggest trying that. I played on a keyboard myself, for the longest time, then switched to keyboard/mouse and never looked back. It's just so much easier to aim with the mouse.
Personally I got used to it over time but I played the original as well (which had very easy beginner missions).

You just have to get used to using the number pad on your keyboard i guess, I've never used a joystick in the game (which I've played on/off since it came out).
I'm having a hard time getting into Freespace due to the controls as well. I loved the X-Wing games and still remember the controls for those, but Freespace uses different ones and has twice as many.

I tried using a Logitech Dual Action gamepad, but I don't feel that I can aim very well with it, and even with 12 buttons the controls don't seem to map in any logical way. I tried using the mouse and found that I can aim much more accurately but can't turn quickly enough.

I'm thinking of using the number pad on the keyboard for turning and the mouse for fine aiming, and not using the gamepad at all.