Q: Can I transfer my ipod music onto a new computer just by plugging in the ipod?

No.  You can transfer your library from the old computer if you still have it:

How to move your iTunes library to a new computer - Apple Support

Otherwise you'll need a third party software to extract the music from the iPod.

I think you cannot, but sometimes, when you plug your iPod in, your computer would read you iPod as a file format, so you may transfer your iPod music onto a computer, while it does not work all the time. So I recommend you'd better use iTunes to transfer your music onto a computer/Mac.

Open iTunes on your old computer.

Connect your iPod to your old computer and wait for it to appear in iTunes.

Select your iPod on the left side of the iTunes window under the Devices section to display your iPod preferences.

Click the Summary tab in the main iTunes window.

Select the "music"and "sync". Then "Apply" at the bottom should be pressed,too.