I used to plan my weekend around not missing Car Talk and Prairie Home Companion. I now can listen to shows like this on the internet so I need to figure out what I don’t want to miss and listen to them at my convenience. I like
Wait Wait, Don’t Tell Me
Car Talk
This American Life
Radio Lab
TED Radio Hour (just discovered this today)
Prairie Home Companion
I forget what else. what do you like I should try?
Hey Jackie, the BBC mostly streams it’s radio output worldwide. Have a look here:
Judging by your taste, the stations you want to look at first are 4, 4 Extra and World Service. It’s late here so give me a shout later if you need to narrow down stuff. 
Michael Feldman’s Whad’ya Know.
Actually, come to think of it, it’s you who has had until recently a not-very-good internet connection? (Sorry if it’s not you)
Have a look at tunein.com - the whole worlds radio is there. It’s an aggregator of every free stream on the planet.
I’ve caught a couple episodes of Ask Me Another and like it; not as witty as Wait, Wait, but it has its moments. Last night there was a round based on The Village People’s YMCA. Jonathon Coulton would sing a verse with hints to some other organization and the contestants had to provide the correct four-letter acronym.
There’s also Says You! It’s not produced by NPR, but many of their stations carry it. It’s also a panel show, but not based on current events and many of the questions have a literary slant to them.
God, I hate that show with the passion of a thousand suns. Complete waste of air time. And yes, I am a supporter of my local NPR station.
I would recommend To The Best Of Our Knowledge to the doper listen list. It’s a serious show about “ideas.”
And A Way With Words because who here doesn’t get excited about the etymology of words and phrases?
Yes, 4 Extra rocks!
But not in the way of playing rock music. More in the way of good radio drama, including Doctor Who on occasion.
I’m listening to BBC World Service right now (The Forum Curiosity) and I will check out tunin.com, thanks.
for internet I have mifi & not enough, what, bandwidth?, to stream video (or much, anyway) but I can listen to radio shows with it.
so much internet, so little time! 
high five
I love Wait Wait, really like Ask Me Another, and listening to Whadya Know for 30 seconds makes me want to drive to Feldman’s house and kick him in the pancreas until he dies.
Anyway, if you like cooking, The Splendid Table is great, and The Dinner Party is always fun - it’s “the show that prepares you to win your next dinner party.” Some episodes of This American Life are riveting, but it needs to be a topic that catches my interest.
To The Best of Our Knowledge looks very good, thanks.
I used to listen to Talk of the Nation and Science Friday, I’m adding those to my list. Also Thom Hartmann, although he isn’t public radio.
What don’t you like about it? I’m fairly fond of it, though the annoying piano jazz interludes always make me change stations for five minutes or whatever. Seriously, why the fuck does being an NPR listener automatically mean I’m also supposed to like shitty modern jazz? Grr.
From the ABC (Australian)
All in the Mind
The Philosopher’s Zone
The Science Show (brilliant, just brilliant)
Look basically anything on ABC channel Radio National is of high quality and you might get an insight into Australia beyond Kangaroos and Mick Dundee!
The Diane Rehm Show is fantastic. It’s a talk show that explores a myriad of topics – everything from best-selling books to current events. She’ll often invite a panel of experts that respectfully discuss their POV, such a rarity today. The audience can ask questions via email, Twitter or phone.
She came to Cincinnati a few weeks ago in conjunction with our local NPR station’s fundraiser, and I went to see her. She answered two of my questions, thank you very much. She’s in her 80s now but just as fabulous as ever.
From the BBC offerings, I listen to Friday Night Comedy from BBC Radio 4, and Mark Kermode and Simon Mayo’s Film Reviews (Radio 5).
Friday Night Comedy has short alternating “seasons” between two shows, The News Quiz (rather like Wait Wait’s quiz parts) and The Now Show, which is more of a mostly-topical stand-up comedy show, including parody songs interspersed during the show.
Kermode and Mayo’s show is film reviews and analysis, plus they usually have an interview with some interesting director or actor.
You really need to stop shrouding your feelings behind wishy-washy language! 
Hey, that’s why there is more than one station on your radio dial. To each his own.
Personally, I find Feldman’s dry sense of humor pretty funny. I will agree with LawMonkey on the short jazz interludes, but the original poster already mentioned Prairie Home Companion as one of the shows he/she wanted to hear. I figure if he/she has the patience to get through the occasional Garrison Keilor gospel duet, a few minutes of jazz will be a cakewalk.
On the BBC you might look for the “Friday Night Comedy” and “Comedy of the Week” podcasts. The former was mentioned above; the latter is a selected episode comedy show from Radio 4 - usually some comedian who has pitched a short series to the Beeb and been given a couple of episodes to try out his concept. Some of these are funny, some are terrible, some are poignant and some surreal - it’s a very mixed bag.
For non-BBC shows I recommend “The Memory Palace” for short and very infrequent but fascinating historical stories. They’re currently part of the Maximum Fun group of shows (which also includes “Judge John Hodgman” and “International Waters”, both of which you may or may not enjoy).
Also seek out “Irrational Public Radio” for a deadly accurate parody of public radio. Weird and wonderful.
And I tried to like “Whadya Know”, I really did, but it was unfunny in the way that sucking chest wounds are unfunny.
September 21, 1936- She’s 76. She sounds older because of her voice problem, spasmodic dysphonia. Her autobiography, aptly named Finding My Voice, is well worth reading. When she was a toddler, her mother used to punish her by not speaking to her, sometimes for three weeks. There’s a short bio of Diane here. A quote:
She is so courteous and civilized and the fact that she struggles with her voice and yet has made a career in radio is proof of major league guts.
I can’t stand Feldman’s voice most of all. He’s imitating (badly) early Johnny Carson mixed with Bob Newhart. Our local NPR station used to carry TWO hours of him on Saturday morning! Yikes, talk about sucking up prime in-the-car-running-errands air time! There was a protest and he was cut back to one hour–still one hour too many IMHO. And I was able to return the unused buckets of tar and intact pillows back to Wal-Mart and get a refund. 
I’m another Feldman hater. What bugs me is that his so-called interviews are always about how “clever” he can be. Just shut up, quit interrupting with your supposed witticisms and let your guest answer your inane questions, already!
Anyhoo, if you like This American Life, you might like The Moth, Snap Judgement, and KCRW’s Strangers.
Another vote for it. That it does a full hour on a topic lets them get into more depth than usual. A few topics don’t interest me, but they talk about some stuff you never hear discussed in the media.