Postgraduate study is open to any graduate with a good degree. In microbiology, there are two main options.

A Master of Science (MSc) is a one year full-time taught course including a programme of lectures, practical classes and, very often, a short lab-based research project (2-3 months long). Some universities offer the chance to study for an MSc over two years on a part-time basis. Many students choose an MSc course to improve their career prospects by gaining skills and knowledge in a specialist area of microbiology or by retraining in a different subject. This can be a big financial commitment since there is virtually no funding available and most students cover their own expenses and course fees. 

A Doctor of Philosophy (PhD) is awarded after completion of a 3 year programme of research supervised by a senior scientist in either a university or research institute. Some PhD programmes last 4 years and include an initial year of taught course and research skills development leading to a Master of Research (MRes). If you are clear that you want a career in research then doing a PhD is the first step. The majority of students apply to advertised PhD projects which are designed by the supervisor and funding includes an annual bursary for the student. It is very rare for a student to come up with their own project unless they are self-funded and can find a suitable PhD supervisor.