Yes. Some flights to places in Florida can be had for less than $100. Certain countries like Thailand or Costa Rica are very cheap as well, providing you can get there. A westerner can live like a King in China too.
Yes, there are less expensive alternatives. I would first recommend picking some place that you can drive to in your car. Then I would suggest seeing well in advance what hotels you can get a discount on.
1) Sign up for cheap airfare alerts. There are a bunch of sites that do this (AirfareWatchdog, Bestfares.com, Fare Compare, etc.), which can help you easily find big flight discounts.
2) Get a travel rewards credit card. Some good starter options are the Bank of America Travel Rewards card and the Capital One Venture card, but you should do some research to figure out which one is best for you. This way, you can start earning points to be able to travel for free.
3) Subscribe to ThePointsGuy's email newsletter and read his Beginner's Guide article on his website (https://thepointsguy.com/guide/beginners). You can learn about ways to game the points/miles system and get updates when travel credit cards with huge sign-up bonuses become available (see if your parents will agree to use your card for purchases they would already be making in order to hit the minimum spend limit).
Traveling can be fun and inexpensive, but you have to know how to shop for deals. My suggestion is to make a list of places you want to go. Then start scouting for deals on places like Groupon and living social. Consider going places off season to see them, so that you can avoid crowds and get a better price. If you are air traveling, be flexible with your dates so that you can fly stand by, and if you plan to drive, carpool with friends so that you all can split gas prices. Decide what you want to do/see and then pick your destinations. If there's a big trip you wanna do, then save up and wait.
Enjoy the world!
There are loads of places to travel inexpensively at any point in your life. There is always a trade off when looking at places to visit however. Some of the places that I have visited that were cheap to get to (low ticket prices), were quite expensive in terms of food and lodging. Other places that I have visited had expensive tickets but were very inexpensive once on the ground.
For inexpensive travel ideas I think Lonely Planets guide books are the best. They offer things to do and places to see that are more directed to backpackers than many other books.
I have had great experiences in South America, Southeast Asia, and Eastern Europe.