Hello Petapan,
Welcome to the Discussion Boards.
First, make sure your iTunes is up-to-date:
http://support.apple.com/kb/HT2365
Second, since you made no mention of any alerts or error messages when you connected your iPhone and synced, and you originally setup your iPhone on another computer, it would lead me to believe that you haven't gone through the Music, Video, Podcasts tabs and selected any of the "Sync" boxes. If you had, you should have encountered an alert telling you that the "iPhone is synced to another iTunes Library" and prompting you to erase and sync with your iTunes Library.
Since the iPhone is designed to work with only one iTunes Library at a time, you will need to let iTunes erase the music/video/podcast/etc. content from your iPhone in order to sync.
Since you indicated that the iPhone mounts in iTunes successfully and iTunes shows it's syncing for a short time, it would indicate that your PC, iTunes and the iPhone are all getting along together and you just need to make the correct selections to sync the content you want on the iPhone.
If you have iTunes purchases on the iPhone that were purchased with your iTunes ID, you will want to choose "Transfer Purchases" from the File Menu while logged into the iTunes Store on the PC with your iTunes ID before syncing any media content from your computer.
If you are going to be syncing contacts with your computer, you will need Outlook 2003, Outlook 2007 or Windows Address Book and you will want to open and setup those applications on the computer before you attempt to sync contacts.
If you are going to be syncing calendars with your computer, you will need Outlook 2003 or Outlook 2007 and have it opened and setup before syncing:
http://support.apple.com/kb/TS1347
These steps should get you up and running. If you encounter any error message, please post the entire error message verbatim for additional assistance.
Charles H.
