Depends what sector you are in. I didn't apply for graduate schemes during my final year (wanted to concentrate on my finals, and wasn't 100% sure what type of role/sector I wanted to commit to). I applied and got an internship before I graduated which I completed in the summer, then did another internship (unpaid which isn't ideal) which I got a job from. I got promoted within 3 weeks - the quickest promotion known in the company.
So I'm a real advocate for internships for both giving you key work experience that impresses, as well as helping you to explore different fields. I didn't have a problem getting an internship as a graduate, but I'm looking at the health/public sector, whereas some companies like large investment banks or marketing firms are very strict about their internships. Even so, you could still find a good internship in a smaller finance or marketing firm (or whatever you're interested in) - there are lots of off-cycle or winter internships not just marketed at students.
Would advise against doing a masters for the sake of it.
Teaching English abroad would be fun and I know lots of people who've had great experiences with it, but make sure you understand how this fits into your wider career plan - it might be worth getting more specific work experience such as an internship in the summer, then teaching abroad come autumn for a while so that you have more relevant work experience when you return.