Well where do you want to start?
You can't go wrong with the Galton and Simpson classics, Hancock's Half Hour and Steptoe and Son both available as cheap DVD box sets (£16 and £24 respectively). Both suffer from "lost episodes" (thanks BBC) but what survives is remarkable. Hancock is an early (back to the 1950s) sitcom with the main characters using their own names (save for Kenneth Williams) but as caricatures, almost grotesques. The Sid James character has elements of his Carry On persona but much darker and more malicious. Steptoe has a particular brand of tragedy comedy. Both have radio versions regularly broadcast on BBC R4extra streamable (and ripable) from the website. Hancock started on the radio. Steptoe along with many other BBC TV comedies had episodes adapted for the radio.
Aside of the general entertainment show The Two Ronnies, Ronnie Barker has a couple of classics with Porridge (with the died too early Richard Beckinsale, father of Kate) and Open All Hours (with David Jason) (dvd box sets for £12 and £10). Just avoid spin offs and remakes!
Beckinsale was also a main support in Leonard Rossiter's Rising Damp (£12 DVD box set) Rossiter also having The Fall and Rise of Reginald Perrin £18 DVD box set.
Going military there's It Ain't Half Hot Mum (£16 DVD box set) and of course Dad's Army (£27 DVD box set) also adapted for radio and regularly on R4Extra.
Going political there is of course Yes (Prime) Minister (£18 DVD box set) and The Thick of It (£13 DVD box set). Can't deny the quality of the latter but the former is timeless, has more subtlety and, worryingly, you still see it's plotlines and contrivances play out in modern politics.
There are a couple of old "Play for Today" comedies, Abigail's Party and Nuts in May both with a young Alison Steadman, more difficult to get but worth tracking down. You can't underestimate the impact of Abigail's Party in particular.
More recently it's hard to go wrong with the Alan Partridge (universe?), a spin off from The Day Today, that started on radio with Knowing Me Knowing You as the highlight (DVD box set containing that and the rest of the BBC material for £10). You either know the monkey tennis reference or you don't. There's also the League of Gentleman (£20 dvd box set) and Little Britain (both started on radio). The latter is a bit harder to track down these days esp at a sensible price.
For more obsure stuff try Ever Decreasing Circles (£11 DVD box set) (Richard Briers as a character with a darker edge than his usual type). The High Life (overpriced though at £24 fot the 6 episode series - there was only one), a early Alan Cumming comedy from BBC Scotland back when they put out the likes of Absolutely and Rab C Nesbit. Or, a personal favourite, A Very Peculiar Practice which has been available as a DVD box set but is difficult to get now. Peter Davison as the idealistic doctor from a mundane general practice arrives at a university mediacal practice expecting it to be wonderful but finds, in the words of Graham Crowden's alcoholic practice chief, "a swamp of fear and loathing". It also has David Troughton (another Dr Who connection - he's Patrick's son) as "Black" Bob Buzzard, ambitious, tech obsessed, sees patients as an impediment to his career and being at the medical practice as a failure and, Barbara Flynn as Dr Rose Marie, the manipulative bisexual feminist ("illness is one of the things that men do to women").
That should be enough for 5 years or so.
