Do a general search on the title of the old show. That should produce a list of sites that claim to have the episodes. Read the rules of any sites carefully before downloading anything.
Stick with some of the established TV sites. The cable channel TVLand has several old shows in rotation on its website. In October of 2011, for example, it had four episodes of "The Dick Van Dyke Show," and seven of "Leave it to Beaver." Hulu.com, which is run by NBC, also has a library of older shows from many networks. The site does ask all visitors to register, and the free registration lasts for just one month. Xfinitytv, which is run by Comcast, claimed to have 27,204 episodes online in October of 2011. Not all of the shows are free. For example, the site's HBO shows were available only to visitors with a subscription to HBO.
Click on the name of the show you want to see. Each site has different prompts, but most of them have simple play, pause and fast forward button functions. Some sites, like Hulu, stream the shows with commercials.
Insert the name of an old show on YouTube. There's a chance someone has posted an episode or two or at least part of an old show. This is a long shot since the production companies that hold the rights to the shows usually get those clips removed.