As you rightly say, most jobs require a degree and a TEFL certificate. The type of degree does not usually matter as long as it's a 3/4 year degree (i.e. not an Associates or Foundation degree). However, a lot of countries require that the degree be taken in an English speaking country so, for example, if you wanted to teach in Korea you'd have to have a degree from the UK, Ireland or one of the other English speaking countries.

As far as the TEFL certificate goes there are many options. CELTA is arguably the top-end TEFL certificate but since you might be looking for work without a degree then it won't make much difference what TEFL cert you get.

This article might help, it's about teaching English without a degree and goes through the options you can follow: http://tinyurl.com/cuhp9kc