You could always look up a 'best of X' to watch the very best episodes, but honestly with ST and DW you're going to miss a lot of the storyline and the personal character arcs that really make these shows what they are.
Instead, maybe watch the show and if you're watching an episode and not digging it, just skip the rest and move on to the next one. You're far less likely to miss important things this way.

First episode of doctor who? Which first episode?

Doctor whos first season of the reboot is pretty rough

House and Dr Who have subjectively boring episodes. Dr Who reboot, for example, have great actors but no so great budget (which is fair for Dr Who as a whole, it's low budget sci fi from the start). But House was quite stable from start to finish, if you didn't like first episodes, you wouldn't like whole show. You may be like the concept of it and some scenes/parts, but it's like picking pepperoni from pizza and eating them separately.

Supernatural is pretty stable in first seasons and get somewhat bonkers later. But it's pretty much character show (same as House and Dr Who) and characters and their chemistry is pretty much always there. So if you don't like what you see now, you wouldn't like it later.

Can't say anything about Start Star Trek.

You can always do what i did with Agents of SHIELD. I didn't like early season procedural nature of the show, it was kinda generic for me. But i kept hearing that later seasons are amazing and i wanted to see Ghost Rider in context. So i watched whole thing at 1.25-1.5 speed, which is helped with parts that was boring to me, but allowed me to not skip character growth as a whole. Funny part: some scenes was actually more natural at 1.25 speed. I guess they could be directed to be longer, so they could help with runtime.

I can't say anything about Start Trek either, I'm more of a Stop Trek kinda guy.

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I feel like if you have to resort to cherry picking episodes, don't bother with the show at all, just stop.

Doctor Who:

I would say there are two routes with Doctor Who. Watch the highest rated episodes so you get a flavour of the kind of stories the show tells and what it's about. Or start a season with a Doctors first episode and watch it through till they regenerate so you get the ongoing character arcs, themes and ideas that are set up for later stories. Some stories work best when you've watched them in season order as the emotion of what the character has gone through resonates with you more. Here are some of the best episodes according to IMDB. I've tried to put only stand alone stories that don't feature or spoil elements of the seasons overall plot.

RTD Era:

• Season 1: Rose, Dalek, 2 part story: The Empty Child & The Doctor Dances

• Season 2: The Girl in the Fireplace, 2 part story: The Impossible Planet & The Satan Pit

• Season 3: 2 part story: Human Nature & The Family of Blood, Blink

• Season 4: 2 part story: Silence in the Library & Forest of the Dead, Midnight, Turn Left

Moffat Era:

• Season 5: The Eleventh Hour, Vincent and the Doctor, 2 part story: The Pandorica Opens & The Big Bang

• Season 7: The Angels Take Manhattan, The Snowman

• Season 8: Listen, Mummy on the Orient Express

• Season 9: Heaven Sent - spoils a character moment but it's a mighty fine episode.

Chibnall Era:

• Season 11: The Woman Who Fell to Earth, Rosa

• Season 12: The Haunting of Villa Diodati

Generally, starting with a Doctors run is a good way to go as well, if you gel with the particular Doctor or companion, then you may find some stories more enjoyable. Start with the first episode of a season and watch it through to the end. Due to a persons taste, episodes will vary in quality in terms of story, but tone, style tend to stay consistent through out a season. You stated you didn't enjoy the first episodes and that's fair enough! Doctor Who often finds it's feet with a new production team every once in a while, so that may account for your experience. Plus, television production and value has changed a lot since 2005 when the show came back and Doctor Who has never had a high budget, so it won't look as polished in places but that's part of the charm for some. If you go down this route, then start in the following places:

*Season 1 (2005)starting Rose for the 9th Doctor (Christopher Eccleston)

*Season 2 (2006)starting with the Christmas Invasion for the 10th Doctor (David Tennat)

*Season 5 (2010) starting with The Eleventh Hour for the 11th Doctor (Matt Smith)

*Season 8 (2014) starting with Deep Breath for the 12th Doctor (Peter Capaldi)

*Season 11 (2018) starting with The Woman Who Fell To Earth with the 13th Doctor (Jodie Whittaker)

Notes: I didn't put any episodes from season 6 as it's quite season plot heavy. Season 10 is brilliant and a personal fave, but isn't rated highly, though I think it's a great season. I prefer RTD's storying telling over Moffat and Chibnalls. Chibnalls seasons are very much back to basics approach to Doctor and I think work best when you've watched the previous Doctor's before hand, though starting with 13 is a good jumping on point as well, I just don't think it's as consistent as other era's. Moffat has 6 seasons as showrunner and so was more experimental with the show and so story quality varies, but there are more good than "bad" episodes. Doctor who has varying levels of story quality depending on the viewer and their like of the Doctor.

Edits made for formatting purposes.

Most of the episodes in both are really good in my opinion.

I'm not very familiar with Dr. Who (I just watched season 1 of the reboot and did not continue), but I can talk about TNG because I have watched the whole show (and I really like it). It's a long show (178 episodes over 7 seasons). Being episodic, each episode is a different story and there are stronger and weaker ones.

Yes, if you don't want to devote much time to it, you can watch the best episodes following whatever guide you want. The episodes are quite standalone, barring the occasional double episode, so you'll be fine sampling the highlights of the show that way.

However, there is something you will miss, because part of the charm of a long, episodic, ensemble show like this is the feeling of living with those characters day to day, so that they become almost members of the family, and following their character arcs. Even weak and campy episodes are enjoyable for me because of that. But, if you don't want to spend all that time, that's a reasonable trade-off you can make.

One thing you should know about TNG is that season 1 and, to a lesser extent, season 2, are considered weaker by most fans. Have you heard the expression "jump the shark" to refer to the moment when a show stops being good? It's an expression that was taken from an incident in the sitcom Happy Days. Well, the opposite expression is "grow a beard", to refer to the moment when a show starts being good. That expression was born from TNG, because during season 2 the character Riker grows a beard that he keeps for the rest of the show, and that's where fans say that the show started being good.

Personally, I also like season 1, although I agree that it has more weak episodes than other seasons.

This is a watch/skip guide. Just somebody's opinion, and I would personally recommend watching more episodes than this list suggests, but you could give it a go if you don’t want to watch many episodes: https://wordmerc.medium.com/star-trek-the-next-generation-watch-skip-guide-e489377bfd9

You could also look the the TNG re-watch at tor.com, and watch only the episodes that are given a 5 or more rating, for example (you can see the rating at the end of the write-up for each episode, just go directly there to avoid the spoilers in the rest of the write-up). It has the advantage that, having a rating, you can set the bar as high or as low as you want. Again, it's just somebody's opinion, and I often don't agree with his ratings, but that's a given when it's something subjective like this: https://www.tor.com/series/star-trek-the-next-generation-rewatch/

With Doctor Who it’s definitely OK to cherry pick, but good luck finding a consensus on which episodes are “objectively good,” (hint: it’s entirely subjective).