Hi there. The first thing is set up your iTunes and get your account in order. If you can copy the entire iTunes folder from the old computer, it should retain all your settings. (including the music, unless you have your music on a external drive). If your music isn't stored in the iTunes folder under MUSIC, either copy it all into iTunes from your previous computer, or use one of the many iCopy softwares to get your library back onto the computer. It's usually under iTunes/Music or copy all your music into a folder and then do File/Add to Library... and import all that music back into iTunes. Then you should be able to just go in and make sure it isn't set to sync on plugging the phone in. Then when you plug it in, right click on the iPhone on the left column and choose backup. That will get your info and apps safely backed up.
At this point you should be able to just sync and everything should be fine. If you don't have the music though, you do have an issue. If you sync it could wipe out your music from your phone. Just check under music for the phone that the same things are checked. Also, Google is your friend. Search, "moving iTunes library from phone to computer" Or just search "copy iTunes to another computer". Mostly, do a little reading before you do it to be sure I haven't forgotten anything and you'll be fine. Good Luck.