Legal research isn't cheap, at least if you're using databases like Westlaw or Heinonline, which will set you back somewhere in the ballpark of $100 a month (although I'm not sure you can even buy access for personal use).
Fortunately, they aren't your only tools available (but you should absolutely use them if they are available to you). Google Scholar is probably the single most valuable tool you have for legal research if you don't have access to a database. So valuable is it, that I was able to write an article using nothing but Google Scholar and have it accepted by a journal (so look forward to that).
Assuming it's not been iced by the time you read this, Google Scholar is a search engine for scholarly articles. It also has an extensive collection of US court opinions. While it doesn't have advanced searches, or helpful summaries like Westlaw or Lexisnexis does, searching for the terms you want will generally get what you're looking for with minimal sifting. Searching for a specific case or article will usually work just fine, although in rare cases it might just give you a citation, which is just a bit useless, really.
Google scholar does its best to find a public version of the article you want, but it may not be able to pull up a .pdf for you. If it doesn't, you can check for alternate versions of the article and you might be lucky and find it on SSRN. You can also try emailing the author and asking for a copy, and they might be able to provide you with some additional insight. If that doesn't work, and the words "shadow library" mean anything to you, well, you already know where to look (but they tend to be a bit less fruitful for law journals, not that I'd know).
If you want UK cases, you can use BAILII. I haven't used it much for research, but it seemed to pull up most cases I threw at it, provided they weren't too old (cases before the 90s seem a bit neglected). There are equivalents for other jurisdictions like CanILII and NZLII for Canada and New Zealand, respectively. There's also WorldLII which lets you search for cases across several international databases.
Once you've found your cases and articles, just use them as you normally would. You don't have to do anything differently.