Will best buy transfer everything from my iphone 4 to the 5s?

I want to buy the iphone 5s from best buy because they'l give me more money for my old iphone, but will they be able to transfer everything like my pictures, contacts, data, to the new iphone?

Yes, they can, but you have to make a backup of it onto iCloud. Once you have a backup and your new phone is activated, you can restore it onto your new iphone. They can help you with that, just make sure you back up all the stuff you want to transfer onto iCloud first.

Its best to back up yourself first to make sure you got everysingle piece of your data!

On Your Phone (Wi-Fi Required)
Settings
iCloud
Then Back-Up from there!

On PC:
Itunes
Top Right press iPhone
Goto summery
Find back-up
And click it

Then your done!

Once you get your new phone, when you set it up remember to eaither press "restore from icloud backup" or "restore from itunes backup"

Pictures, would have to be done on your own through iTunes on your computer (excluding pics from the camera roll, photo stream)

Everything else you could also do on your own via a backup again using iTunes.

Yes not a problem, all you need to do is back up all your date to your apple account