Access to journals for individuals outside of an institution?
As a preamble: maintaining a publishing system has a cost. You may shift it to the authors, or to the reader, or have it financed directly by governments or supranational organizations, but there will always be a cost. It's true that in the current system, with a mixture of private publishers and non-profit ones (academies, learned societies, some well-administered professional societies), some players may be making a bit more money than we wish. Academic publishing has not yet fully adapted to the digital age, and it will get leaner by doing so. However, we have to recognize that this will always have a cost.

Now, answering your first question: there is a growing trend to go towards open access journals (and open access options in many other journals), which shift the cost onto the authors. Moreover, some funding agencies (for example NIH) have a public access policy, meaning that they ensure that publications stemming from research they fund are publicly accessible in some form (in the case of NIH, the final version of the preprint must be made available on PubMed). Now, different fields of research are moving toward this new paradigm at various speed (biology is faster, chemistry is not moving at all). In the current state of things, it must be noted that apart from a few renowned exceptions (PLOS One, for example), most open access journals are far from prestigious.

In some fields, the publication system has hoped earlier on the digital bandwagon, and it is routine for authors to publish only preprints of their work (and publishers have copyright transfer terms allowing it). For example, in mathematics and physics, arXiv.org is a very popular source of preprints. In some other fields, most journals don't allow preprints to be posted at all…



Regarding your second question, there are a few options for you to gain access to some paywalled journals:


Public library: if you live in a big city, the local public library likely subscribes to a few major scientific journals: Nature, Science, things like that.
Get a guest reader status at the closest university: some institutions allow access to outsiders in their library, either by not actually checking who gets in (as long as you behave), or in some cases by actually having an official “guest reader” status (which may require some paperwork or approval).
Membership in a professional society: individual membership in the American Chemical Society gives you access to their journals at a rather decent price.
Writing to the authors: if you see a paper you don't have access to, just write to the authors asking for a PDF.
Reddit Scholar :)