My favorite is Are You Being Served.

I loved The Vicar of Dibley with Dawn French as the lead actress. Funny.

Allo Allo is nothing but catch phrases.
Hardly comedy gold.
Peace be your pie.

You should try Allo Allo. I don't know why it is so unpopular here, but it is one of the best british stitcoms. I have watched it dozen of times and still cracks me up.

first belive me,please. try Only Fools and Horses
http://www.imdb.com/board/10081912/?ref_=tt_ov_inf
i promise,best comediy ever!

If you love Aussie comedy like that, you should check out 'All Aussie Adventures'. Comedy gold, my friend lol
Peace be your pie.

raps
It's not a Brit-com, but if you can get a previously owned DVD copy of "Mother and Son", it would be worth the money. I haven't seen it on our local PBS station for at least 20 years now.
This winner comes from "down under" [Australia] and it is one of the funnier 30 minutes shows you'll ever see.

I can't believe nobody mentioned the two great Ronnie Barker ones
Porridge
Open All Hours
Sure, they're a bit old these days, but they are still classic funny. Very well done.
And how could you go past
The Goodies
The Thin Blue Line
Going straight (sequel to Porridge)
Hancock's Half Hour
Men Behaving Badly
There are so many to choose from. In fact, I'm kind of jealous if the op. It's like being a kid in a candy store really. I would love to be able to watch all these for the first time.

If you liked the original Office and Extras
You should check out:
Peep Show
The Trip
The Thick of It
Don't watch anything else before you watch all of those. Those are the only ones you need to see.
But regarding Black Adder and Red Dwarf. Black Adder isn't that great, but it's a classic and it has a cult following and everybody loves it (but it's not that great.) Red Dwarf is a highly clever scifi comedy that usually has very interesting/complicated plots that are pretty cool to watch, BUT the characters are EXTREMELY annoying, so you'll have to overlook how annoying the characters are to enjoy the very neato scifi plots. I'd say Red Dwarf over Black Adder (though, Red Dwarf has more episodes, so it will take longer to watch.)

PHONESHOP! I seriously think Channel 4 is the HBO of Britain I laughed SO hard at this.. I might even prefer it to Peep Show
http://www.channel4.com/programmes/phoneshop

Here's my little list for you:
The Good Life
Are You Being Served
As Time Goes By
Vicar of Dibley
Outnumbered
Nika
I've got a
Ruby Red
Desire
Like a Virus
Like My Last Hope

Father Ted is great, as long as you don't mind a bit of 'silly' humour, as there are no high-brow references or anything. I personally find it hilarious- youtube 'My Lovely Horse,' if you like it I think you'll appreciate the show's humour.
I'd recommend Yes Minister/ Yes Prime Minister, an 'The Thick of It' if you like political humour. The former is based on earlier British governments, and the latter more recent governments but both make fun of the fact that Civil Servants basically run the political system in Britain. I personally prefer to watch 'The Thick of It' (with the utmost respect for the show which spawned it) mostly because I think Peter Capaldi as a constantly swearing Scottish man is perfect.
You mention you've watched 'The Office,' so you might want to check out Gervais' more recent offering 'Extras.' He also has a new show only beginning to air on British TV called 'Derek' so you could keep an eye out for that.
Blackadder is a must, as is anything Monty Python.
QI is great, and quite educational too as far as TV programmes go! Try 'A little bit of Fry and Laurie' too, very funny.
More recent British comedies that are worth watching are Outnumbered, (the kids often are unscripted, makes for some very funny dialogue) Black Books, the IT Squad, Miranda, Twenty Twelve (based on the planning for the 2012 Olympic games, I found it really funny), Peep Show (a must).
If you're having trouble finding ways of watching some of these, I know that a lot of them are posted on Youtube. (Usually under the format 'Twenty Twelve (for example), s01 e03)
This mess is gonna get raw, like sushi, so haters to the left

The Inbetweeners, Absolutely Fabulous, Outnumbered and Life's Too Short are some good ones to start with.

Peep Show
Lead Balloon
Pete vs Life
Stressed Eddie
The Trip

My list of 'Greats' -
http://www.imdb.com/list/8ifRqu1j2iY/

Finally someone mentioned the greatest britcom of all time - Father Ted, must see British Comedy, even if I feel 2 mins isn't really giving it a chance! Mind you that review was from someone who thinks 'cash in the attic' is a comedy (WTF?).
Not to offend our neighbours I should point out that Father Ted was written by and starred Irishmen, but I think it still falls under 'britcom'.
Do a search on you tube for 'Father Ted explains perspective' for a flavour.
In order of quality (top to bottom):
Father Ted
Fawlty Towers
Yes Minister
The Inbetweeners (TV show not movie)
Peep Show (seasons 1-4 only)
Blackadder (seasons 2 & 3 only)

I check in here now and again for ideas of what to watch, these days Netflix
has MANY British shows online,more than my memory can recall,but includes.
.
Falwty Towers
Monty Python
Are You Being Served
Green Wing
IT Crowd
Hyperdrive
The Worst Week of My Life
All Creatures Great and Small
Monarch of the Glen
Black Books
Top Gear
The Avengers
Mr. Bean-The Whole Bean
Torchwood
A bit of Fry and Laurie
Dr. Who (some older Eps)
Little Briton
Doc Martin
Hotel Babylon
Father Ted
Waiting For God
The last two: I actually turned off Father Ted after 2 minutes, made it
through one episode of "Waiting" but didn't like the characters at all.
.
They also have "Cash in the Attic", to slow and repetitive-it's a 30 minute
show stretched to an hour,and the false drama is tiringooooh, will she get
700 pounds for the party,oh dear !Another lady wanted 800 for a vacation,but
had like a half-million pounds of crystal and glass in her garage.blah.

The Inbetweeners
is hilarious. Season 3 is a bit of a let down though. A movie has come out which follows the 3rd season and is fantastic.
This is my murder-swagger, I wear the skin-trophy of my prey.

My girlfriend is Canadian, and the first sitcom I showed her was I'm Alan Partridge. She absolutely loves it. (I once got her a mug from Snappy Snaps for her birthday. Alan's face on one side, and a cat on the other.)
So I imagine you'll appreciate it, being a fellow seal clubber.
Also, Spaced is great. She didn't like that nearly as much, but that's because she's a girl, so didn't ANY of the references.

I'm an American who has recently started watching British comedies, and here are the ones I have enjoyed the most so far:
Peep Show - This show is brilliant! I would rank it up there with Arrested Development as one of the greatest sitcoms of all time.
Pulling - I wish that there were more than just two seasons of this well-written and funny show.
Saxondale - I heard that there is an American version of this show in the works, but I can't imagine it working nearly as well as the original. The actor playing the main character is excellent.
The next shows on my list to watch are Mighty Boosh, Green Wing, and Spaced. Unfortunately, I haven't been able to find a way to watch The In-betweeners, but I've heard good things.

I am in a similar boat to the OP.
Loved Spaced. (watched doctor who forever and watched red dwarf previously)
Didn't get into Fawlty Towers and Blackadder (especially fawlty towers)
Checked out Black Books (funny by not as funny as spaced)
Heard about IT crowd
Watched Garth Morenghi's Darkplaces
Watched the Mighty Boosh
As a cautionary tale:
Try Spaced/Black Books and the IT crowd.

Peep Show
Spaced
Black Books
The IT Crowd

Start with "Fawlty Towers".
"Pain is an old friend,as long as it is here all is fine,but when it is gone,that is when I worry."

Ab Fab is genius. I'd also say Only Fools and Horses, The Smoking room, Peep Show, Misfits and Inbetweeners.