Hmm... In general, I don't watch a lot of old t.v. series. Through most of it's history it has been a lesser form of storytelling, at least in the U.S.A.. There are some real stand outs though, particularly in the comedy genre.
Star Trek - is overall a great series with interesting concepts, good acting, and is very forward looking and ahead of it's time. There are some really silly filler episodes in the run, but most of those are at least still fun to watch.
I Love Lucy - Truely earns it's spot as a classic. I don't think I've ever seen an episode that didn't make me laugh, and a few are all time classics. It's amazing how well this show holds up, it's one of the oldest I've ever seen and it's better on writing and production values than most of the shows that came after for decades.
Three Stooges shorts - These are great physical comedy shorts, and they are a lot more clever than most give them credit for. It's not a series you'd want to binge, but every now and then I do love watching one of these.
Looney Tunes shorts - some of the best animation of the era, and all time classic comedy skits.
Super Dimension Fortress Macross - Anime bucks the trend with t.v. shows, they are usually the equal to movie and OVA releases. So I won't list every anime t.v. show that I like, just this one. Because the original Macross really is something special. I don't want to say too much, you really should watch this show. It's a war story about humans struggling to survive an alien invasion aboard the last capital ship in the fleet. It's dramatic, tragic, uplifting, and thrilling. The enemies in the story are not generic bad guys, and as things develop it gets complicated. It also has a fairly long coda that tells the story of what happens after the big climatic events. That's a story you rarely see, and it's not a clean feel good ending, but it also has tons of hope.
That's pretty much it for stuff before I was born (1982). I do want to see the Twilight Zone, I've heard lots of good things about it. I'll say less about the shows after I was born becuase most are very well known.
Batman: the animated series, Gargoyles, Animaniacs, Old Simpsons, Seinfeld, Scrubs, Malcom in the Middle, Star Trek TNG, these are show I come back to again and again. One show that I truely love that has been forgotten by most is Titus. The original creator uploaded the entire series to Youtube years ago, you can watch it here:
Titus T.V. show. It's a comedy series that can sometimes deals with hard dramatic themes and stories. Like Scrubs it's mostly focused on the laughs, but when it gets serious it can really hit you. It's also absolutely hilarious, my family still quotes this show all the time.