I have all my contacts saved on my SIM-card but I want to move them to the phone which is Galaxy note so that when I change the SIM-card my contacts are still saved and I don't need to save them all again. Is there any way to do this?
Yes, just go to:
Contacts -> Menu -> Import/Export -> From SIM to Phone
Or you can view this more detailed article to help you: http://androidadvices.com/import-sim-contacts-android-phone/
When you are in contacts you have to edit your contact list by importing or exporting contacts. There's nothing on the screen to indicate how to do this but pull up the menu by pressing the icon on the very bottom left of the phone (on the note3 this is not on the screen but on the frame around the screen) and then you can click on import/export and move contacts to either device or sim. Card by Just going down the list and clicking on all the contacts you want to import or export.