If I buy another iPhone, will my sim's contract continue on that phone?

I'm looking to buy a new iPhone (I currently have a 5s with 16GB). Long story short, I've had 32/64GB on every Apple device I've owned as I take plenty of photos/videos and having 16GB simply isn't enough.
I've got a contract for my iPhone (unlimited data/texts, 500 minutes) which I want to keep.
I want to buy another iPhone (with 64GB), so if I put my sim inside the new iPhone, will my contract continue on the new phone and it would be that simple? Or?

You answered your own question in the title: Your SIM card's contract won't stick to the PHONE

That totally depends upon your contract and if you need to have some good security app, install Leo Privacy Guard. It hides images and videos and it also locks various applications which can be very useful to us