Should I apply for internships, travel overseas and experience a different culture, or just stay on campus and enroll in summer classes? What would a successful student do?
If you'd like to have fun, stick around, take a few classes and party. You could graduate earlier, but into what? What happens after that?
If you'd like to jump into a job at a high paying, well known company after graduation, you better start looking for internships with high paying, well known companies.
If you'd like to be able to tell people you've been to another country, go to another country. I don't say experience, because I personally think that takes more time than a typical vacation and involves getting outside what you read about in the travel books. It's up to you, but many people just want to be able to say they travel, and have been outside the country, so if that's you-do it. Believe it or not, if you want to experience somewhere cool and interesting, sometimes you just have to travel just outside your hometown. You could join half the USA and fly to a foreign country and stay in a hotel. You could also get to know all the hiking or mountain biking trails, or raftable rivers in the region you live. I like to think there are great experiences all around, just have to spend the time to look for them.
Personally, I've done all three. Spent two years living and working in a foreign country, interned one summer with a fairly well known tech startup, and worked two summers on campus. I'm with a large, international company working full time now. They all ended up kind of connecting, as one experience generally helped lead me towards another.
Starting a business while working full or part time would make for a very interesting and potentially worthwhile experience as well.
If you frame this question in terms of where you want to be (i.e. at a startup, at a big company, in grad school, married or no, in a different part of the world, in debt vs no debt), it will help you identify experiences that can help you get there.
My #1 recommendation: Build something lots of people can use. You need a portfolio of things you've built outside of school/work.
You can do this while you intern somewhere, go abroad, or take extra classes in school (meh).
My personal preference would be to go abroad to a country where your own currency goes very very far, so you can live/travel cheaply and also hack while you are at it.