1st to 2nd shift (Automatic)
 

I noticed when redlining it in 1st gear and shifting to second in manual mode (didn't really pay attention to automatic mode) it feels like the rear wiggles a little causing me to slightly correct with the steering wheel

is this normal due to the power of this car even in automatic..i know on a 6-speed you can get the tire chirp

I don't really remember my 2013 base 370z doing this back then...then again it was 3 years ago

Have u had the differential bushing replaced yet?

I got it used with 8000K miles and I haven't touched it so I'm not sure if it was replaced or not yet

First thing I'd do is get the rear suspension inspected for worn/broken parts - repair/replace as needed - and then get a good alignment.

I had an alignment done recently

But I did have a nail puncture in my rear driver side tire that was patched recently. I'm unsure if it did this before or not because I don't redline it too often otherwise I'm flying over the speed limit

Yes it's normal. It's extremely easy to get the Z's rear end to "break loose" due to it's power and lack of weight over the rear axle.

I can get the rear of my Z to "wiggle" from a stop just by flooring it in full automatic.

I also feel it wiggle alittle when flooring it from a dead stop

I can definitely get it to wiggle when powering out hard through a corner in 1st gear