Luggage comes in all shapes and sizes. You can get a suitcase with two wheels, you can get a suitcase with four wheels. They have hard cases, and some have soft cases. You can also go for a backpack with or without straps and carabineers.
Most people will have one purse/cross body bag or backpack (often considered a personal item), one carry-on sized wheeled suitcase, and one larger wheeled suitcase. This applies for both long and short-term programs. Just because you're going twice as long doesn't mean you need twice as much. If you're flying, be careful about the weight of your checked and carry-on luggage (read more about that in How to Pack it All, but be aware that different airlines have different restrictions). You may also want to bring an empty duffle bag in case you have more clothes and souvenirs to bring home. You can also use it for shorter overnight or weekend excursions.
Overall, it's your choice! Play around with different options, and consider what combination is going to be best for you and how you plan to travel. You should choose your luggage based on what type of packer and traveler you are. Take into consideration how easy it will be to drag your luggage around with you through airports, on local transportation, and up and down stairs...elevators aren't always readily available.
There are a ton of places to buy luggage!
Our office (and the University of Kentucky) don't have agreements with and are not sponsored by these places, but if you were to catch our staff on the street and ask where to buy luggage, these might be some of their answers: