Lenerod,
In your case, sync to your iTunes, and make sure your backup for your iPhone 4 is created, you can check this by clicking in the Sidebar (Ctrl-S to show/hide it) on your iPhone and in the upper right side it should say Last Backup Completed: today's date.
Then you plug in your iPhone5s (Assuming you have never sync'd this phone before) and it will ask if you want to replace a different phone or set up as new. Choose replace and select the backup you just created. Now everything that was on the old phone should be on the new phone, including contacts.
If you are already syncing the iPhone 5s, then try some of the other suggestions above. Like the link just above your post.