Learn the basic process of what you need to do when using an iPhone with a new computer.
Syncing Your iPhone
The iPhone really requires a form of centralism where it is constantly tied to your base computer. If you are switching computers, such as to a 
new computer, you will want to transfer your iPhone over with you. Using your iPhone on a new computer is fairly easy once you go through the basic set up functions and know how to navigate through these issues. Here is a tutorial on the basic process of using an iPhone on a new computer.
Set Up Your New Computer
The first thing you have to do with your new computer before you sync your iPhone is to get it where you want it to be in terms of drivers and software. Make sure to download the newest version of iTunes, or you can use the CD that you may have from when you purchased your iPhone. You are going to want to make sure that everything is up to date before you even try to plug in. If you want to transfer any music or video that you may find on your iPhone then you will want to physically transfer those files from your old computer onto your new computer. The easiest way to do this is likely to just add that media with the rest of your data to an external hard drive when switching over between computers. This way you can update all the media that was on your iTunes account on your old computer to your new computer.
Plugging Your iPhone to The New Computer
When you do plug in your iPhone you will be given the option of syncing to that computer. You will take this option and go through a series of basic steps to reassign your iPhone to this new computer. This is incredibly simple, and since the media that is on your iPhone will already be in the iTunes of your new computer when you do the first sync you will not have to bother with that process. What you will have to do, however, is transfer over your App Store items to the new iTunes account. This process is similar to the process that occurs when you purchase iPhone games or applications from the iTunes App Store using the App Store application on your iPhone. Then plug the iPhone back into the computer. iTunes recognizes that there are purchases on your iPhone that are not also present on your iTunes account. You will want to update this so that all those iPhone applications, both free and costly, are then backed up on your iTunes account. This will not take long and should be done during the first reassignment sync process. You can also match any podcast subscriptions you already have, especially since you will be using the same iTunes account that you have used all along, just a different actual iTunes’ program and computer. Your iPhone podcast subscriptions should likely remain the same, but just double check in the proper podcast location. The sync for the podcasts should continue to remain standard, but if not you can easily resubscribe to the podcast.