This is intended as a general purpose question. I'm not looking so much for specific information about any particular place (although if you do know any titbits about Sheffield, UK, then I'd love to hear it) but instead about how anybody with a desire to know more about the place that they live can find it out.
What are your methods?
What resources should I look for? Are there any resources that I wouldn't expect (i.e. things that aren't books, e.g. oral histories, newspaper archives, town archives)
Where/how can I find them?
How much detail can you reasonably expect to find about your local area (e.g. the city you live in, or maybe even a particular part of the city).
What are some top tips, that may not be obvious to a novice inquirer?
Thanks in advance
The first question is how far back are you interested in going? If it's <100-150 years, you can often find newspaper archives and other similar items. The older things are usually housed in special archives (because they're old), but sometimes you can find scans online or in a local library or university library.
I'm not familiar with the UK system, but in the US we have a system of "Historical Preservation Societies", with people who specialize in preserving and educating people about local history. I'm unable to find much in the way of UK institutions that are similar, but you might be able to inquire with the Sheffield and District Family History Society (http://www.sheffieldfhs.org.uk/) and perhaps they have some resources that could help you. Alternatively, you may be able to reach out to some of the people involved with the UK Historical Association Sheffield Branch (http://www.history.org.uk/resources/he_resource_1171.html) and perhaps they have access to sources or archives that might be of interest to you.
Most towns/cities (at least in the US, and presumably the UK as well) have a State or Town Library in which all the important records for the city (such as various governmental decisions, family records, etc) are kept. Again, I'm not sure how the UK does it, but perhaps there is a library at the place of local government in your area that you can contact for more information. This may be the best place to start. A quick Google search brought up the Sheffield City Council Library's page on Archives and Local Studies: https://www.sheffield.gov.uk/libraries/archives-and-local-studies.html
I think that's the best place to start for you.