Because some people get the air date and story date confused when I mention this I feel the need to differentiate the two.
Air Date refers to the real life day in which the episode first aired on TV. In this case that would be October 29, 1966.
Story Date refers to the time in which the characters have traveled to within the context of the show. In this case that would be March 6, 1986. Though the wiki says December for some reason. I remember watching it and making a note about the date because they actually said my birthday.
That being said, the day I was born the Cybermen were invading Earth for the first time and for some reason despite multiple invasions we keep forgetting about it. I mean it's possible I could of just noted the year and imagined that the month and day were there as well but still I would of been a baby within the Doctor Who universe despite the fact that in real life my parents didn't even know each other yet. Time travel is fun and confusing.
So what was going on in the Doctor Who universe the year you were born. Keeping in mind that this is by Story Date not Air Date. You can also use that wiki to look up episodes by when the story actually takes place. Just look for your year and it'll give you a list of Doctor Who stories that take place within that year.
When I looked up 1986 I also got "The Locker Room", "Mondas Passing", "The Nightmare Fair", "Old Boys", "Time Wake", "The Way Forwards", and "Whatever Happened to Iris Wildthyme" but the Tenth Planet was the only TV story.
It would be kinda interesting to put all the episodes together in chronological order by the Story date and watch them that way. Though that might get kinda confusing because you'd then be watching the Doctor's life in the wrong order. But I think it'd be interesting to watch Doctor Who starting with the Tenth Planet and go Chronologically just to see what all has happened on Earth in my life time according to Doctor Who. I'm just kinda curious exactly how many times since I was born that the Doctor stopped an alien invasion and exactly which incarnations were there.
I can't think a better way to get into a show than to pretend it's all actually happening.