Can this phone work in other carriers like AT&T or even abroad?

Verizon vs. AT&T
Unfortunately, Verizon and AT&T phones and networks don't really like each other. Verizon uses a CDMA (Code Division Multiple Access) network where each cell phone (or wireless device) is permitted access to their network individually. With CDMA you cannot move from the intended service provider to another with the same device.

AT&T (as well as most of the rest of the world) on the other hand use GSM (Global System for Mobiles) networks. With GSM networks rather than authorizing a single phone to be used on the network they authorize a SIM card to be used. The SIM card can easily be taken out of one phone and put into another and that's all you have to do to change phones or with an unlocked phone, change networks (more on that later).

Due to the different types of networks you cannot go from one to the other though.

U.S. Phone vs. Overseas
As I said above, AT&T and most other countries use GSM networks, where switching your SIM card can easily be done to change the phone you use or the network you're on. That would bee optimal if you plan on doing a lot of international travel, but there's a catch. You have to make sure you get an "unlocked" GSM cell phone, because most of the phones you buy in the U.S. are locked to a single carrier (see oligopoly) which prevents the owner of that phone from using it on any other network despite the fact that is how the technology was designed to work. This is mostly because carriers pay for most of the phone (typically around $800 for "top of the line") and require the buyer to pay a smaller portion and sign a 2 year contract. If you get an unlocked GSM phone you will pay significantly more for the phone, but you won't have a contract and you can switch from AT&T to Europe Mobile (I just assume that's the name of a company) with the switch of a SIM card and you will no longer be roaming internationally. You will have to use prepaid SIM cards for whatever country you're in at the time, so also keep that in mind. Then when you come back to America, replace your AT&T SIM card, re-enable your account (unless you want to pay for AT&T while you're gone and not using it) and continue using the same phone.
NOTE: Right now Best Buy (online) has an unlocked AT&T S4 (refurbished) for $690. So you can get the same phone, but also if you're paying that much out of pocket I strongly recommend getting the Best Buy protection (it's the best insurance I've found) unless you're willing to pay the original cost again if something happens to your phone.

Other tips and tricks
Constantly using prepaid SIM cards can get expensive even if it is significantly cheaper than international roaming. If you have a good internet connection consider using Google Voice when you're at wherever you're staying so you can call any number in America for free from anywhere in the world, and calling a number in any other country is usually less than ¢3 per minute.