New phone

See this support link from Apple-->Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

And, as razmee says, you can just switch SIM cards (and you should), but this will only switch your phone number, it won't move anything else.

Assuming you're using the same phone carrier, yes your number will activate on the new one, if you're talking about your app data, pics, message etc no that is not store on the sim card.

You would need to restore from your icloud or itunes backup