This is a great question, Russell. For me, growing up in The States made it difficult to find a station that aired 'Doctor Who' with any regularity. In the beginning, when I would play with that second knob on the TV (yes, I'm that
old... when you had TWO knobs on the TV for channel-changing, and had to get up off the couch or chair to change channels, and you had three small knobs for adjusting volume, brightness, and contrast... again having to get up to do so - there were also hidden knobs inside a pop-open panel that adults only were allowed to touch
for adjusting Vertical Hold, Horizontal Hold, Tint, Color (now known as "Saturation", or how intense the color was), and two more adjustments my old brain just can't remember)... where was I? Oh yes...
In the beginning when I would play with that second knob - the UHF channel knob (which my mother would have a fit over), I first came across 'Doctor Who' (though I never caught the opening, so didn't know what it was called), it was in the early 1970s, about 1973, I think. This obscure UHF channel (PBS) was showing the Jon Pertwee stories. To me, it was just this tall guy with greyish curly hair fighting off these robot-like things (I would later learn they were Daleks), and he had this magic blue box he traveled in. (At the time, I thought it was just the entrance to his spaceship.) I had trouble finding that station on a regular basis, so my Pertwee stories were few and far between, but I was captivated by it all.
Later in the 1970s (I was in high school, so it was after 1975), I found a commercial station showing this intriguing show consistently on Saturday mornings at 10 o'clock, only now it was some guy with brown curly hair and a huge scarf, but with the same blue box, and he was fighting off those robot-like things. This is when I learned the show was 'Doctor Who', those monsterbots were Daleks, and that blue box - the TARDIS - was the entire ship and bigger on the inside. I thought I was captivated before; now I was mesmerised!
The problem was, with the exception of that UHF channel (which I never found again), that commercial station only ran the Tom Baker stories from "Robot" to "Underworld", and then would cycle back and repeat the airings. It wasn't until 1986 when I was stationed on Governors Island in New York harbor that I was able to get 'Doctor Who' (in feature-format: a complete story in one sitting with no breaks between episodes) from a nearby PBS station in New Jersey (and oddly enough, it was a UHF channel again!). Again they were only running the Tom Baker stories, only this time they ran from "Robot" to "The Invasion Of Time" before cycling back. And then it happened...
During one of their pledge drives, they started airing the William Hartnell stories from the beginning! So every Friday night from 9 PM until the story ended (it was the last thing the station showed before shutting down for the night, so they could air a 4-parter for about 100 minutes - a little over 1 1/2 hours, or a 6-parter for over 2 1/2 hours). For longer stories, like 7-parters, they would break things up and show what amounted to 4 parts this week, and the last 3 parts next week. Patrick Troughton's "The War Games" was 5 parts this week, and 5 parts next week. But still, whether they broke up the longer stories or not, they showed every story. So I got to see every available episode from "An Unearthly Child" to "The Greatest Show In The Galaxy"... actually almost every episode; they didn't show the black-and-white Pertwee episodes from the color stories, so Part 3 (or 4?) of "Planet Of The Daleks" and Part 1 of "Invasion Of The Dinosaurs" never aired over here, which sort of broke the story for "Planet Of Daleks", and made one wonder what was going on at the start of Part 2 of "Dinosaurs". But aside from those two, we did get every available episode. (I was transferred to a "No 'Who' area" before Season 26 aired in the US, so it was a while before I got to see Sylvester McCoy's final season... in the 2000s when we got the DVDs from Netflix.)
So I got to see every episode in order. (At the time there were only FIVE of Patrick Troughton's stories available: "The Dominators", "The Mind Robber", "The Krotons", "Seeds Of Death", and "The War Games". My, how things have gotten better over the decades!) Then of course came the 1996 TV Movie with Paul McGann, and the hopes for a revival were sadly dashed. When the New Series started in 2005 we watched it like glue. (Still do.)
Netflix was a great source for all of the New Series episodes, and a handful of Classic episodes, and after watching everything in order (because there were no DVD releases then, and any VHS, if you could find it, was expensive), when we got the urge to watch something, we would pick and choose, and yes, that even included "the clunkers". (And then the BBC pulled everything 'Who'-related off of Netflix. They've got things on Britbox now, but honestly, when it comes to Classic Who, I think my DVD collection has more stories than Britbox! They do have all New Series episodes, though - which is good because I've fallen behind with collecting the New Series box sets.)
So we still pick and choose, and watch whatever strikes our fancy at the time. (Since we've seen them all now, we can jump around without worrying that we'll miss something.) As for watching the same story or episode over and over again, at least for the wife and I, it doesn't diminish the impact. Take "Blink" for example. Since we haven't gotten Series 3 on DVD yet, we've got "Blink" saved on our DVR (from BBC America). No matter how many times we watch it, it's still just as creepy! (Angel statues always spooked me before that episodes aired. Now? As we said in The Bronx where I grew up, "Fugyet abou' dit!" I'm pushing the big "Six-Oh" and I can't turn my back on an angel statue!)
So we watch whatever looks good to us at the time. "The Three Doctors" never gets old. Neither does "Blink". "The Masque Of Mandragora" is also a constant favorite, as is "Pyramids Of Mars". "The Eleventh Hour" ranks up there as well. "The End Of The World" is also a great one for the New Series. So, basically we've got loads of favorites from the Classic and New Series that we watch over and over. And there are "the clunkers" that we watch as well. Some of the more dark, sad, depressing episodes we'll go long stretches before watching again because, well, they're dark, sad, and depressing ("The Angels Take Manhattan", "Earthshock", "The Hand Of Fear" - it's the departure of Sarah Jane, "Resurrection Of The Daleks" - it's the way Tegan leaves, "Doomsday" - that one makes my wife cry... probably a dozen more depressing ones if I took the time to think of them all).
It's a roll of the dice as to which story/episode we watch, but after watching them in their intended order once all available stories aired in the US, we just watch what we want. (The same with the original 'Trek'; like 'Doctor Who', those episodes you can watch over and over again; anything from TNG on, once you've seen it, it's a pick-and-choose... though there are some original 'Trek' episodes we go long stretches with for the same reason: dark, sad, depressing... "Balance Of Terror" is one; but "The Trouble With Tribbles" never gets old, and neither does its 30th Anniversary sequel, the DS9 episode "Trials And Tribble-ations", and of course for great action there is "The Doomsday Machine".)
Anyway, that's how the wife and I watch 'Doctor Who' in the off-season: what strikes our fancy at the time be it Classic or New. When the new episodes are airing, well those take precedence, of course (and since we DVR them to blast through the adverts, if it's a good new episode, we'll save it and watch it again during the week while we wait for the next episode). And this was probably more than you were ever expecting, I expect.