Archives, archives and more archives. Many have digital services and such. Additionally for English speakers websites such as JSTOR are good for secondary sources.

Can you be more specific what topics you're interested in and what you're trying to answer?

I'll quite often come across a question on /r/askhistorians that I want to "research", which often starts with consulting a good reference encyclopedia like The Encyclopaedia of Islam or a comprehensive topic-based reference like the Cambridge Histories. If I don't find what I'm looking for directly from that I'll then look at what references those works cited and look for those books, monographs, or articles. If that still doesn't turn anything up, then I must be dealing with an especially difficult question, but I might then turn to looking for specialist sources outside of my field or to combing through primary source documents that might turn something up.

This can be where a non-professional with limited access might run into problems. The most recent example I can think of for myself was that I was looking for anything that could explain the specifics of where, when, and how the border between Iraq and Jordan was formed. I know that I can probably find the information I'm looking for in Patricia Troye's Palestine boundaries: 1833-1947, or in collections of archival materials like Documents on British Foreign Policy, 1919-1939 (including some of which if I recall correctly have been published in specialized collections on the Middle East). And I know that that the answer is in a particular State Department publication that I once found reference to.

But I don't have ready access to any of these materials and so the answer to my question remains a bit of a mystery to me.

For a non-specialist, the best way to do any of this would probably be a trip to the library. Librarians are generally quite good at helping you to find the books you're after. It is, in fact, generally their job :)

As to /u/Doppelagent's point about archives, I think that's probably true if you want to write and publish professional quality "historical research." You're probably not doing new "research" if you aren't engaging with some sort of primary source materials. These need not necessarily be actually in an archive. Some government archives are published and available elsewhere (the Documents on British Foreign Policy volume that I referred to being such an example.) There are of course other kinds of primary sources as well.

But if you're just looking to learn more about topics that interest you this almost certainly isn't necessary to do.