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.