Getting Music NOT purchased from iTunes, from an old computer to new computer
There are (at least) two simple ways of achieving this:


Select the songs in iTunes and drag them to a folder in the Finder. Copy said folder to the other computer (via WiFi, external drive, Disk Target Mode, whatever). Drag files out of folder and drop on iTunes.
iTunes Match. For a cost of $25 per year, you can store all your music in iCloud. Whether you bought said music from iTunes, or ripped it from CD, or got it from anywhere. Then you can access it from all your devices - iPhones, iPads, Macs, iPod Touches, AppleTVs - anything that can see iCloud.