You hate it when people push you to watch stuff without solid reasoning?
Fuck it I have nothing better to do, LET'S DO THIS.
Doctor Who is a pretty good show. It's definitely far from perfect, pretty hit and miss actually. When it hits, it's usually really solid though.
Synopsis: Mad man with a time/space travelling box that's bigger on the inside goes on adventures and stops aliens from destroying shit. Usually the Earth. He likes the Earth, his very few friends are there. Also he just likes humans, we're weird and contradictory and stupid and lovely. He's also the last of his race (he kind of... killed the rest of them. Stopped a war that would have destroyed the universe. Tough break.)
He also has a habit of picking up semirandom humans (usually women) and taking them on said crazy adventures as his companions. Usually one at a time.
Oh and when something happens that would kill him, he regenerates into a new body. All the memories are still there, but the body changes, and so do some preferences and the personality. (This is how they change actors.)
There have been 12 Doctors (or 13, depending on semantics, but let's just call it 12). One through seven happened back in the old incarnation of the show from the 60's to the 80's. Number eight was in the 90's movie that was almost universally panned.
The show came back to TV in 2005 with Chris Eccleston (might remember him as the Invisible Man from the TV show Heroes), who is best described as PTSD Doctor. He's fresh from killing his whole race, he hates everything, mostly himself, and season 1 is pretty much him regaining his faith in humanity and universe as a whole. He's my favorite because he's just so intense, but in the rare times that he's happy it just shines from every pore.
Seasons 2-4 (and some movie-ish things between season 4 and 5) brought David Tennant as the Tenth Doctor, and he's the most widely known and loved in the fandom. You may also know him from the BBC show Broadchurch, or as Barty Crouch Jr. in Harry Potter and the Goblet of Fire (the asshole who was impersonating Moody and had the weird tongue thing). Tenth Doctor was a puppy. He had no reservations emotionally. When he was happy, he was over the moon. When he was sad, he was just crushed. When he was angry... Dear god just back away. Fans just adored him, and with good reason- I just happen to like Eccleston a bit more.
Seasons 5 through 7 brought Matt Smith as the Eleventh Doctor. He's just... He's just weird. He looks alien, he acts alien, he's just an oddball. This is when DW started really hitting it off in the USA, so he's probably the most well known Doctor outside of the fans. He does bring a kind of wacky charm to the role though. He's the reason so many whovians get all excited about bow ties and fezzes.
Finally, the most recent season (8) brings Peter Capaldi as Doctor #12. You may know him from his large acting career- and if you don't, you should set to fixing that. He's a much broodier Doctor, more of a crotchety grandfather than previous. It's more of a return to center really- Doctors 9, 10, and 11 were all super young compared to the 60's-80's doctors. Heck, the first doctor ever was like 70.
Anyway, you wanted to know solid reasons why to watch it. I'd just have to say that on the whole, it's a well done show. It has its faults and bad episodes, but most shows do. Heck, even Star Trek: TNG has its "Justice" and "Symbiosis..."
For me, the show is made by the characters. We meet these characters and get to know and love them through their trials and tribulations. I'll never forget the feeling of meeting Rose Tyler, hating her, and slowly warming up to her as she matures and starts thinking about more important things, while at the same time she acts as a healing balm for the Doctor. Don't get me wrong- I love the monsters (most of them... we can ignore the Slitheen and the Absorbaloff), but what keeps me coming back is the characters. No matter how hard the weeping angels tried, no monster can engage me more than the joy and suffering of the Doctor and his companions. I love the show for its methods of exploring people and the world.
If you wanted to give it a try, I'd suggest that you first just ignore everything you've ever heard about it being the BEST SHOW EVAR, and then watch "Vincent and the Doctor," "Midnight," and "Blink." These are three of my favorite stories, and are the most likely to give you a fair idea of whether or not the show is for you. They all combine the "monster of the week" with the characterization I love the show for.
Always feel free to ask if you have any questions about it, as you can tell I can spout off more than I care to admit on the subject.
Heck, maybe we could add one of the above episodes to the agenda along with CAH and GotG!