Oh dear, I certainly couldn't go and watch like thirty or forty years of old episodes, lol. I would've been very leery about venturing at all if this thread didn't reassure me I don't need to.
My sister tried to get me to watch with David Tennant and I just had such a hard time, but I feel she probably picked like a really poor choice of episode, lol. She showed me this two-parter that looked like it was some sort of a climax to a long story or something, like the whole episode was so very intense and I didn't have any clue at all what was going on, and it was like cheesier than James Kirk's Star Trek, lol. I never watched Doctor Who again after that, it just so totally turned me off, but I'm sure she loved the episode but just didn't understand she didn't pick a good one for a newbie.
I don't know if I'd even be all that interested if we weren't getting a woman Doctor, I feel it's going to be very fresh and probably a very wonderful place for someone like me to start watching.
Did the two-parter that you were shown contain a character named Rose?
DW now has 13 different Doctors, one of them is bound to appeal to most people somewhere. The trite response being "Tom Baker!!!!!" but there's generally a good reason for that.
When it comes to arc style shows, a lot of episodes people think are ooey-gooey-squee-dee-great are the ones that require the viewer to know and like the characters' mannerisms. Classic DW, to compare, almost never had arcs so - for the most part - one could show any story out of the blue. Like "City of Death", "The Seeds of Death", or "The Aztecs". Would I recommend to a newbie anything like "The Caves of Androzani"? No way. Why? Because one has to know and like the characters, which means picking a dozen Davison episodes and that includes "Planet of Fire" since Peri is introduced and it's important to know a bit about her setup, FWIW. Or "The Edge of Destruction", the Doctor himself has a character arc in his very first three stories (spoiler: it's about his personal growth, having landed on Earth after escaping numerous harrowing adventures and being paranoid of humans and calling them savages. Ian and Barbara are the focal point, who turn him around. Even in "The Daleks" there are clear signs of the Doctor being more complex and moral than some of what his first story presents.)
That and Kirk's cheese tastes good!
