If a DOI or other identifier is included, you can click on it to find an online copy of the article. This may or may not be free to access, but will give you a place to start. If the article does not appear free to access, you may still be able to find the article elsewhere, whether online or through a nearby library. Consider the resources in the following points as further guides to accessing such articles.
Install the Unpaywall extension on your browser: you will have a 50% chance of finding the full text in open access, wherever you found it.
Search for the article title on Google Scholar. If the initial result is behind a paywall, try clicking on the "All X versions" link - this will tell you if other databases include this article, and may help you find an open version. From here, you may be able to find additional sources on similar topics by clicking either the "Related Articles" or "Cited By" links appearing under most article's link in the results. Articles found using these links and may provide you with information to expand your search.
Using either the DOI, Google Scholar, or the journal's website, find out what databases index the article in full text. You can then see if either your local library or the Wikipedia Library provides access to these databases.
Use WorldCat to see if your local library has a physical version of the journal.
Request the article or the journal through your library's interlibrary loan service, if available.
Reach out to the author(s) of the research paper by email and ask them for a copy.
Note that websites like Sci-Hub offer free and direct access to academic journal articles, but there are legal questions about their use and neither the Wikimedia Foundation nor the Wikipedia community endorses them.
Google Books will often give access to a few pages or a snippet view.
See if other editions are available (although the content or pagination may differ).
Use WorldCat to see if your local library has a physical version of the book.
Request the book through your library's interlibrary loan service, if available.
Leverage your contacts with people studying or working in higher education facilities to get access of master's and doctoral theses from those institutions.
If you only need a chapter in a collective work or fragments of a research thesis, reach out to the author(s) of the work by email and ask them for a copy.
Note that websites like Library Genesis offer free and direct access to books, but there are legal questions about their use and neither the Wikimedia Foundation nor the Wikipedia community endorses them.
Newspaper articles
If possible, search a quote from the article to see if it has been republished elsewhere.
See if either your local library or TWL provides access to the newspaper or to a database that indexes it in full text.
If this is an online newspaper and you see a paywall, try archiving the webpage with the article you look for in archive.today or in the Internet Archive.
See if an archived version of the article is available via a search feature on the newspaper's website.
Use WorldCat to see if your local library has a physical (print or microfilm) version of the newspaper issue containing the article.
Request the article or the newspaper through your library's interlibrary loan service, if available.
Get help on-wiki
NOTE. Before seeking other volunteers' help, you should exhaust all other possibilities to access the content yourself. In particular, check if the Wikipedia Library or your institution offers access to your desired content.