How to activate the sim in my iphone 4s into a iphone 5S?

The phone that i'm using is the iphone 4s, I wanted to buy the iphone 5S (the phone itself), get a new sim and stay on my contract. I know that they two iphones don't use the same sim. So after i buy the iphone 5S and the sim what do i do next? Do i just have to activate it? Do i put the sim in myself? My carrier said they wouldn't put the sim in for me because i didn't buy it from the store.

I originally had the galaxy note 2 which had a micro sim. I traded it for an iPhone 5c and cut the micro sim from the micro sim to a nano sim with an cutter i ordered off of amazon. The iPhone 5s uses nano sim along with the 5c. My phone works great and I did not have to activate a new sim I recommend doing this method because you skip the whole activating process. If you don't want to do this method take your phone to a local phone shop and pay the guy to activate the new sim. Hope this works!