Great question.
I do love classic Star Trek. I also haven’t dived too much into British Sci Fi but I did really enjoy the old British TV series of Hitchhikers Guide to the Galaxy.
I actually don’t watch a ton of TV even with streaming. Most of my tastes don’t align with my wife’s. I do have HBO max. I don’t have britbox, but used to. I’ll have to look into the classic who era, that sounds like my cup of tea.
I gotta rep for Inferno, which is one of my favorite Pertwee stories and features a cool riff on the “evil parallel universe” concept as well as a very tense story.
For the 6th and 8th Doctors, I will say they both have much longer runs with much better writing in the Big Finish audio dramas, but you need to be willing to listen to audio dramas. All the Big Finish stuff is pretty good, but it’s great that 8 gets so much more than one movie and 6 actually gets some good writing.
As a side note, my favorite Big Finish stories are the Doctor Who: Unbound line, which imagines several alternate universe Doctors (including Derek Jacobi and David Warner), although I guess that’s not so much a good starting point as recommendation for fans. For fans of the Hinchcliffe era, he’s written several 4th Doctor and Leela stories (with Tom Baker and Louise Jameson reprising their roles) as the “Phillip Hinchcliffe Presents” line.
The Hitchhiker’s Guide TV series was produced by a lot of the same people as were doing Doctor Who at the time (and the not-yet-5th-Doctor Peter Davison appears as The Dish of the Day in the Restaurant at the End of the Universe part) and it was right around the time Douglas Adams was the head writer for Doctor Who (he procrastinated on both by working on the other).
That’s so facinating! I’ll be looking into this. Thank you
As well as getting to see the moustacheless Brigadier (sorry, Brigade Leader) completely lose it for once, and one of the funnier denouements from any series, yeah, this is a really good call too. If push came to shove and I had to name only one, I think the Brig is probably my favorite Companion out of the entire run of the show, even though he actually traveled with the Doctor only, what, two or three times? And that’s some supremely stiff competition.
“I’m not FROM your world!”
“Good. Then you won’t feel the bullets when we shoot you.”
I would echo those who recommend starting with Eccleston and working your way through New Who. The stories are excellent and the actors playing the Doctor are generally extremely good at what they do. Some later seasons do suffer from poor writing, unfortunately, but not enough to dim my enjoyment of the show overall. For older Who; honestly you can easily pick and choose. I find any serial, but especially ones with The Master, The Daleks, or the Cybermen to be well worth it, as well as any you can see from right before or right after a regeneration. It’s also worthwhile to go all the way back to the beginning and start with the First Doctor’s first serial to get a decent grounding in the BBC’s original ideas for the show. Also, Tom Baker in general is a delight and I highly recommend any of his serials.
ETA: I should mention my all time favorite Doctor is David Tennant (number Ten), and some of his episodes are easily among the best of new Who, in particular “Blink”, which introduces the Weeping Angels and has made me nervous around statues for all time. Tom Baker is a close second though, and as I already stated, virtually all of his serials are well worth it.
I’m coming to Colin Baker’s defense to say that the Sixth Doctor episodes are nowhere near as bad as people make them out as being. It probably helped that I went in expecting the worst and I didn’t start with The Twin Dilemma. While the actual story in that serial is fairly decent, I can see how he might have given a bad first impression. Of the Sixth Doctor serials I watched (all of them except The Two Doctors and the one with Rani), while there are a few that are mediocre, none of them struck me as flat out bad. I suspect the real reason for the Colin Baker hate aside from the above mentioned bad first impression is that his was the one where the reviled John Nathan-Turner (with his fondness of clothes with loud colors) put his stamp on most strongly.
A little update for everyone.
I first watched The Eleventh Hour I dug the chemistry between the doctor and his companion, and the backstory of her being a little girl waiting for the doctor. Although, I will say the whole “Basically… run” was super cheesy and corny. Although, I feel like that was my impression, because the doctor still seems like a quack and is randomly running around with a stick that glows and “fixing things”.
I think decided to go back to classic Doctor to see if I would enjoy it more. Got a 7 day trial of BritBox and started with the first doctor, first episode and LOVED it. It reminded me of a Twilight Zone Episode mixed with classic Star Trek, and also the doctor reminded me of Willy Wonka, in that he was extremely intelligent, but came off wacky to those who didn’t understand him. I also loved that this doctor just didn’t care about the humans at all, and his candor and sarcasm was fantastic.
I feel like Classic Who is going to be my jam. The New Who is fine, and maybe I will grow in my taste for it, but for some reason New Who comes off really cheesy, while Classic who is like a well aged wine, could also be because I grew up watching classic sci-fi with my dad, but we never watched classic who.
Anyways, thank you all for your input and taking the time to lay out suggestions and explanations.
Revelation of the Daleks is some perfectly sound Colin Baker, I think.
Arguments can be made for Vengeance on Varos and Mindwarp (featuring BRIAN BLESSED)
Fun aside, the Doctor Who forum I help moderate, has a setting where anytime someone mentions BRIAN BLESSED it’s always converted to all caps. 
I am always happen to see BRIAN BELESSED turn up and bring his bellowing energy to anything, and, when pressed, I do what is considered an OK imitation of his voice.
Series 4 is the strongest of the Russell T Davies years
Season 5 is the strongest of the Steven Moffat years
But honestly, just hitting the highlights of the modern series will be fun too!
I started with Matt Smith and Karen Gillan on season 5. I thought it was the perfect starting point. New Doctor, new companions, new show runner. I watched until I finished the latest season, then went back and started over with Eccleston on season 1.
I still think 5 is a great entry point.
I would start with Tom Baker for the classic shows and if you are not going back that far, Season one with Billie Piper and Christopher Eccleston. 
Would co-sign on RotD. Rewatched that recently and did enjoy it, notwithstanding the slightly shonky Styrofoam obelisk used prominently at a crucial point. Oh, and Alexi Sayle’s whole performance*.
I suppose my objective judgement is clouded here because Baker was the Doctor when I first watched the programme, so I have an outsized love of his seasons and McCoy’s. So in short, don’t listen to me on this one!
*If anyone doesn’t know who this is I guarantee it’ll be obvious if you watch the eps.
I am constitutionally incapable of hating on an episode wherein Alexei Sayle blows up a Dalek with a beam of concentrated rock and roll. That checks far too many of my boxes.
I am in that minority whose favorite Doctor is the Seventh, especially once he dumped Mel and got Ace – alongside the Second and Ninth. Granted, the competition between all fourteen of them is mostly decided by digits waaaay to the right of the decimal point, so it’s a near thing. 