Once you understand where the money for science and publishing comes from, you will understand that there is no real shame in trying to access papers that are behind a paywall. Essentially, I’m going to recommend that you move from legitimate to illegitimate means, and then if all else fails to get an interlibrary loan. This is last on the list as someone will have to pay for it and it will take some time. Your library may have an allocation that they can use before the bill comes to your advisor’s desk.
Unsurprisingly, this should be your first port of call. It’s probable that your library has an electronic portal that you can use to access all of their subscriptions. But don’t forget that they have stacks with old copies and you might need to physically enter the building. Try it. They are great places!
Google Scholar has lots of links (on the right) labelled with “[pdf],” and these are often integrated with your library. If there’s nothing at first, click on the “All 9 versions” link (below and to the right). This may reveal that there is more than one version with a pdf, and you might have access to one of the others. Check for a link to the author on google scholar, and then look for a link to their website.
Sometimes, Google scholar won’t show a pdf, but you can Google the title “in quotes” and get back a direct link to the pdf. Although rare, it’s worth a try before moving on.
Authors often have official university websites, and sometimes their own home curated sites that may have pdfs of all their papers. Use the clues at your disposal, including a general search engine, and follow the trail.
Most university libraries curate an institutional repository of pdfs of published final manuscripts accepted for publication, so-called Green Open Access copies. These are identical to the printed versions save for the typesetting and proof reading that is done after this accepted copy.
Some authors will respond very quickly to reprint requests, while you won’t hear from others for sometime, or at all. However, this was the traditional way of obtaining a copy that you didn’t have access to, and is definitely worth trying. You should be able to get the email address of the corresponding author from the journal website. Sometimes these are hidden and you need to jump through hoops to find them. You could also search for the email addresses of any co-authors, and also write to them.
Many academics also have accounts on academic social media sites (e.g. Academia.edu; ResearchGate.net). You will need to join the network (using an email address from an academic institution) and have your own account, but these house a lot of pdfs that you can’t get elsewhere.
There are a few of these, and some have plugins to web browsers. You may need to be away from campus and using your own machine, or in an internet cafe to access them (a lot are blocked by universities through pressure from publishers). For some, like Sci-hub, you will need the DOI (you can get this from the publisher’s website). At the time of writing, Sci-hub has >85,483,812 papers. Unpaywall.org has >28,112,995 papers. Library Genious or Libgen claim to have 3 million files. Open Access Button don’t say how many they can reach, but “millions” of papers. And freefullpdf is a website that searches the internet and indicates where you can download pdfs (but sometimes you might hit a paywall).
The chances are that you have friends at other universities where the subscription you are looking for is not behind a paywall. If not, then ask your advisor or labmates, or post it on your lab’s (electronic) notice board.
Librarians have contacts at other libraries. Some have had careers at other institutions or have met people at conferences who’ll be prepared to help them. Librarians are great people and well worth getting to know. Their job is to help you get stuff, so it’s always worth engaging with them.
This comes last on my list as it costs money. If what you are after is a book, then you might end up going here very quickly as many older books are not available online. If it is a paper, it will likely be copied or scanned into a pdf where it occurs. If it is a physical loan, the interlibrary loan will be time limited. When filling out the form, make sure you know where the loan will be billed to before you submit it. And don’t come here first, the chances are that you can get it somewhere in the list above much faster and cheaper.