This depends on the model iPhone you wish to trade in. Models more than a few years old will not likely be accepted for trade-ins. You can check your eligibility through the Apple Store app or website. The value of your trade-in will depend on the model and condition. If you sign up at an Apple Store your trade-in value will be applied to your purchase. If you enroll online you will receive a gift card.
iPhone Upgrade Program
Yes, you can keep both your carrier and your same service plan if you get your iPhone through the iPhone Upgrade Program. However, if for some reason your current plan does not include a data package, you will have to add one. Also, corporate and prepaid service plans are not eligible, so if your service is paid for by your employer you won't be able purchase an iPhone through the Upgrade Program and keep that plan. See here for more information on joining Apple's iPhone Upgrade Program.
iPhones bought through Apple's iPhone Upgrade Program will be unlocked so they can be used on any network. When you buy your iPhone, Apple will set it up to work on your carrier's network (AT&T, Verizon, Sprint, T-Mobile), but you can switch your carrier at any time without having to unlock your iPhone. See here for more information on joining the iPhone Upgrade Program.
Yes, you can buy multiple iPhones through Apple's iPhone Upgrade Program. The only downside is that if you make your purchase online (rather than at a brick-and-mortar Apple Store), you'll need to start a new transaction for each iPhone you want to purchase through the program. See here for more information on joining the iPhone Upgrade Program.