A few teen shows from the late 1950s remain classic, such as, Father Knows Best, The Donna Reed Show, Leave it to Beaver.
In the mid-60's, TV was youth oriented, largely because of The Beatles. They seemed to have slapped corporate America silly with youth, as in teen or anyone under 30. Ads called for the "Now" generation, the "In crowd". As the world focused on youth and young adults, TV teen shows appeared, the first being "Gidget" with Sally Fields as a teen, with teen problems. Its still fun to watch-core issues of teens NEVER change. Others would follow, like, My Three Sons.
However, some of the best teen shows are in a school setting, usually high school, because of the variety of interaction and issues involved. Since 1990, there have been many TV series in this setting, all excellent shows that are mostly "clean drama" (no vulgar words or explicit sex), with excellent actors\actresses in lead roles. Unless you have a tween or a teen in the family, you are likely to miss them, discounting them as "stupid". Nothing could not be farther from the truth. All are VERY entertaining as a show and the issues teens face today (as if a parent does now know!).
The best Jr. High School show remains Degrassi High, done in the mid-80s. It is a Canadian production that is serious drama about what this age group faces in terms of problems. The show is a bit dated, yet, the issues in the episodes are still valid. The acting is fair, as these were REAL students at the school. Another tween show of worth is Zoey 101,with Britney Spears' sister in the lead role. Of course, the mother of all teen shows running nearly 10 years is, 90210 Beverly Hills. It captured teen drama in the 1990s. of course by 1995-6, the characters were now experiencing "college" level problems. By 1999, they were young adults, married and done.
In 1994, another great edgy teen show started, My So Called Life, a teen girl facing issues with parents, friends, drugs, boyfriends, sex, driving. Casting and acting are the best. Tight scripts that we all relate to regardless of age in some way. It was on TV for two years. In 1999, TV had two great teen drama series, Freaks and Geeks, and Popular. The first one had the teen drama setting in the early 1980s at a small HS. You had cliques from the freaks (long hairs, grunge look) and geeks (the idiots, in appearance). The starring girl wears a military jacket and hangs out with hippies, yet she is really straight. Her brother is such a geek, it is hilarious. Typical sibling problems arise, typical parental issue. Guaranteed to bring back memories. It lasted only one year. Popular, has the familiar HS setting of today, like 90210, the characters are well developed and interesting, the two teen girls have the lead roles are from opposite cliques and when their parents get married, now they are family. It covers a host of teen topics seldom covered like gays, discrimination and all the usual ones. The series ended in 2001.
By 2003-06, teen shows developed into the best one called Veronica Mars. Here, the lead role has a teen who is also a private eye. Teens come to her with issues that she helps resolve. Her dad is also a private eye. The whole series is really solving a murder with many subplots. The soundtrack is cutting edge, the dialog is so spot on, acting is superb. The year 2004 produced the first all teen boy issue drama called Life As We Know It. It is very much like the 1994 version for girls. From 2003-06, OC (Orange County) was a very popular teen drama set in a Los Angeles HS. It was popular for it used the formula that 90210 did. The last one, Gossip Girl, 2007-2011, concerns itself with rich teens facing off in school, sort of like "Mean Girls" movie and Popular.
Any of the mentioned TV shows are on DVD. Any of them are worth checking out, especially if you have a teen in the house.