Glasgow uni is meant to be very good for classics, I'm applying there this year.
Becoming a classics teacher is a hard slog, it's not as simple as becoming a teacher of a more mainstream subject.
To teach, you must have an undergraduate degree in something classics-related. Then, you need to do a specialist teacher-training course, which at the moment is only offered at Cambridge and Kings College. Between these two unis there are about 20 places in the whole of the UK. Sussex uni has introduced a teacher-training which would qualify you for classics and history teaching.
After this, you need to find a job. There are few schools which teach classics, and fewer which will have openings.