You can't stay on after your course finishes. Time spent in the UK on a student visa does not count toward any residency or citizenship applications.
There are fiance visas, but these are for non EU citizens outside the UK that allows them to come to the UK to marry. You can't apply for this inside the UK.
Unless you marry before your student visa expires you will have to leave the UK on completion of your course. Please be aware that being married to a British citizen does not give you the right to live in the UK, you would have to apply for a spouse visa after marrying. That is another extremely difficult process. If there is any hint of you marrying to stay in the UK your application will be denied.
Look here: http://www.ukba.homeoffice.gov.uk/visas-…
Over recent years the UK has been tightening visa rules for non-EU citizens. Work visas are extremely difficult to get, you can only apply for jobs on the skills shortage list, these are mostly in the medical, scientific and engineering fields and all require higher education and qualifications and extensive experience. As a new graduate you would not qualify.