If I buy a new iPhone, can I take the SIM card out of my old iPhone and use that?

I just want to upgrade to a 5S (from a 4S) . My plan is on Straight Talk & I want to purchase the phone from the actual online Apple store.

Uh, Nope. Why? 5 has a smaller sim card. You can get another one from ST for free or even cheap to use with it. Not a problem. See this list. Best luck.
http://support.apple.com/...-us/HT5554

You will need a new SIM card

Sure you can

Make sure the phone is unlocked and you can just swap the SIM from the StraightTalk to the new iPhone 5S. You'll also need th trim the SIM as iPhone 4S takes microSIM and iPhone 5S takes nanoSIM. If you need iPhone 5S factory unlocked cheaper than retail, mail me.