I'm wondering if anyone can give me some advice for my problem. I have a 2002 I35 w/ ~170k miles.

It doesn't seem to affect driving but most of the time when I start my car and put it in either reverse or drive, after about 5 feet of movement, I hear a somewhat loud noise. It sounds as if a spring is being sprung if that makes any sense. This only happens after the car has been off for several hours.
It is not the rear brakes, I've heard that before, and it definitely sounds like it's coming from near my feet. This is why I say the origin is the transmission but I could be wrong about that.

I was just wondering if this is normal or something I should get checked out...?

Other problems that may or may not be related with the transmission are low freq. vibrations when coming to a stop and sometimes shifts can be quite hard.

 

There's a sound that my car makes rather often after I move a couple of feet but I always thought it sounded like a motor whirring, a pump sound or something. I could imagine someone describing it as a springy sound as well so I'm curious about whether it's the same sound. If it is then I'm fairly certain that it's normal (though I don't know what it is).

 

I could see it being described that way. It's good to hear that I'm not the only one hearing this and that it's probably normal.