Don’t worry, Dr Who is often called just ‘Who’ by the fans, so you’ve no worries in the naming front.
Even though I have drawn comic strips for the NZ Dr Who Fan Club magazine, and even though I have a friend who is utterly obsessed with the show, I still couldn’t tell you which ones of the episodes to see - there are hundreds of the things. So I’ll look 'em up.
Each story is actually (usually) four half hour episodes put together. There have been Eight official Doctors so far, not counting (ooh, what a giveaway) the Valyard.
I would recommend watching the first Doctor’s episodes, as he established them. That’s William Hartnell. There aren’t many of his eps around now. Try ‘The War Machines’.
Second Doctor, played by Patrick Troughton, had some fun ones. Anything with Jamie and Zoe as companions should be a giggle. ‘The Tomb of the Cybermen’
Third Doctor, Jon Pertwee. He was a quirky, esoteric Doc, and most of his cases were earth bound. Get one involving UNIT, with the Brigadier and Liz Shaw. ‘The Daemons’ has a classic line in it.
Fourth Doctor - Tom Baker. Perhaps the most famous, definitely the longest term. Funnier than the rest. A good episode is ‘Ark In Space’ which is frigging scary. Stay away from any K9 or Sarah Jane Smith episodes.
Fifth Doctor is Peter Davison. I always liked him. Even if he was kind of distracted a lot of the time. Had an interesting time with a companion called Adric, who died, and Harry, who betrayed him. See ‘The Five Doctors’ even though it’s not that good.
Sixth Doctor. Bit of a mess this one. Played by Colin Baker, his time was ruined by an annoying busty pseudo-American companion named Peri, and a mess of an ending to his era with the ‘Trial of a Time Lord’ debacle. See ‘Revelation of the Daleks’.
Seventh Doctor - my favourite - played by Sylvester McCoy. Not popular with many fans as he was a bit goofy, and had worst companion of them all, Mel. But then got best companion of them all, Ace. This Doc’s intention was, as the middle Doc, sets up some things that his other incarnations had to encounter. See ‘Battlefield’ or ‘Survival’.
Eighth Doctor was the singular movie Doc played by Paul McGann. Watch and be amazed at the mess they made of the potential franchise.
There. Hope that helps.

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