There are many web sites devoted to popular TV shows. Broadcasters usually have the 'official' sites, but with a few notable exceptions, the 'unofficial' web sites produced by fans are (in my humble opinion) far superior. Tabulated below are links to sites for the shows I am currently enjoying. This isn't, by any means, a comprehensive list, but they all have their own links, and the usual internet search engines can produce many more.
Two sites in particular have caught my attention: Al Garn's Garn's Guides, and the twin sites of Epguides and TV Tome. These both have episode lists for hundreds of TV shows, with cast lists and dates of first broadcast. Epguides and TV Tome are, I believe, now working together, and have very similar information, and cross-link to each other. They also have brief programme synopses for each episode of over a hundred shows!
For various reasons, episodic programmes like these are often not shown in the production order - especially for the first transmission. Beware, then, that Al Garn's lists are in the order of first transmission, anywhere in the World (usually the USA, Canada, or the UK). Repeat transmissions, or transmissions in other countries, are therefore more likely to be in the order intended by the production company. The Epguides/TV Tome lists include (where known) the production code for each episode, and are usually listed in that order.