Google Scholar will automatically let you know if you can get access to the paper directly from the computer you're using. After that there is going to a library (they often have subscriptions to at least some journals) and visiting author websites.
Also, if you already know which papers you need, you can always ask here on LW
In addition to going directly to articles, consider dropping an email or two to researchers working on those topics (perhaps once you've found an interesting article of theirs). Many are very willing to provide their overview of the area and point you to interesting resources. While there are times when you won't get a response (for example, before conference season or at the end of the semester), most are genuinely pleased to be contacted by people interested in the topics they care about.
If you're looking for articles in the life sciences check out: http://www.hubmed.org
For physics and math check out: http://arxiv.org
If you were looking for articles in ACM or IEEE's digital libraries you could maybe direct-message a sympathetic LessWronger who pays for subscriptions and is willing to slightly bend copyright law and send you a copy. There's bound to be at least one.
I second the recommendation for Google Scholar, it indexes multiple versions of each paper and these often include freely available PDFs.
So I'm all geared up and ready to go trawl through some academic journals so I can figure out how to tackle a problem or two, and I've realized I completely forgot how to go about it. Not going to a university with paid subscriptions to JSTOR, etc. doesn't help either. So where exactly does one go to access such journals? Is there a cheap way to do this? Thanks.