Well, if you liked Rome (I've been watching it myself recently, I'm a few episodes into season 2), I can recommend some good historical dramas. If you want more of Rome, you can go for I Claudius - it's fine with age, and is littered with good actors from WAY back when - BRIAN BLESSED, John Hurt is a fantastic Caligula, and it has Patrick Stewart WITH HAIR. More recent historical pieces would include the Borgias (second season coming up, Jeremy Irons is great as Pope Alexander VI (Rodrigo Borgia)).
Along the sci-fi line, Star Trek's been mentioned and it's definitely good - best place to start is probably TNG. If you want something more recent, or darker and story arc driven, you could try the new Battlestar Galactica. And of course, there's always Doctor Who - like any show, it can be hit and miss at times, but when it hits, it HITS, and if you want to see an example of that, find 'The Waters of Mars', damn that's good.
As for British humour, I'm only going to make one suggestion so as to drive it home well: BLACKADDER. Watch it. Season 1 is different than the others, so try starting on Season 4 or 3 first, but it is good. Rowan Atkinson being CLEVER and funny, Stephen Fry, Hugh Laurie, top of their game.
Other good stuff? Game of Thrones is excellent, a thoroughly medieval tone in a fantasy world where the fantasy isn't stealing the stage, but the intrigue, setting, and characters are. If you want a more modern political drama, try the West Wing - great ensemble, stellar writing, and still relevant to today's politics in almost every single way (More low tech. That's pretty much it) despite it ending almost a decade ago. You want something that's all about characters and character development? Watch Season 1 of LOST, then keep going if you want to follow the plot further. The list goes on, but that should be enough to get you started, I think.