Transfering Iphone music to a new computer


This may be a easy process and Im just being paranoid lol but in order to transfer songs from a Iphone to a new computer on Itunes, All I need to do is just connect my iphone to my new computer through USB, open Itunes, and it should transfer every song and playlist that's on my Iphone , correct?

I ran into issues in the past with Itunes and syncing from computer to phone with having a mixture of purchased Itunes songs, songs copied from old CDS, and downloaded songs from mousebits and such. The issues mostly occurred when trying to transfer songs from Itunes on my computer to my phone and vise versa. For example if i updated my playlists on my computer with mixture of Itunes purchased songs and songs that I had downloaded from, lets say mousebits, and then went to sync, It ended up being a huge cluster **** where songs would not sync properly, would not transfer over at all, playlists would be deleted, etc. Basically it was a huge challenge keeping both Itunes on my computer and my phone identical. And if you attempt to change the settings from automatically transfer to manual, which I usually like to do it manually, then it threatens to erase your library and stuff. Huge massive headache lol. So much so that I basically gave up for awhile even bothering using Itunes on my computer and syncing my songs. Unfortunately now Im finding I need to update my playlists on both my computer and my phone with songs that I had downloaded off of Itunes and such.

I personally would have no problem switching to a new application instead of Itunes if it made syncing a lot easier, but then that would create a issue of the songs that I had purchased from Itunes?

Thanks in advance to anyone that can help me out.