For me the most elusive has been a British bitter.
Things foreign brewers usually get wrong on British beers :
The water is way more mineralised than they think - calcium should be 100ppm as an absolute minimum, don't be afraid of sulphate at 400ppm or more (at least for northern styles, you might go as low as 250ppm SO4 down south. See eg
the bottom of this page from one of the main UK lab service providers, although the header is shifted one column to the left).
Serving temperature - below cellar temperature (55F) it just kills the flavour, I like to serve at about 50-52F so that it comes up through the sweet spot as I drink a pint.
Overcarbonation - British beers are a delicate balance between all the different components, and too much CO2 just screws up the balance. And cask/bottle carbonation just gives you a finer quality of bubble (qv champagne).
Too much crystal - I'd regard Fuller's as on the crystal heavy side, and they only use
7.2% of light crystal in their main beers. Northern beers tend to use half that, and eg Boddington's (the real one, not the export version you guys see)
has no crystal at all. For some reason foreign brewers think that there should be way more crystal and the result is a gloopy mess. On a related note :
Be aware that British brewers routinely colour their beers with brewer's caramel, so don't add speciality malts just to hit a colour target. Flavour is way more important than colour, you should never add flavour ingredients for colour purposes.
Use British ingredients, in particular British crystal malt is not the same as US caramel malt.
BU:GU - generally in the 0.7-0.9 range, towards the lower end in the South, towards the higher end up north (0.85 would be my personal taste)
Those are the musts. As a general pointer, I'd start with a best of 4.2% ABV as about the sweetspot - sub 4% bitters are much harder to do really well.
And although I have a massive personal bias in favour of Goldings over Fuggles, I would still argue that a 100% Goldings beer is probably the best place to start for newbies - they're classic, consistent and should be easy to get hold of. Then you can start experimenting - First Gold would be a good second choice, personally I love a bit of Bramling Cross, Bullion, or Jester mixed in with my Goldings.
Do be generous with your hops, and do dry hop (as British brewers/publicans have been doing for at least 200 years). Hops here generally come in 100g packets and I tend to just use a pack in a 20 litre batch (ie 5g/l or 3.3oz in 5 US gallons) although it is a little extravagant.
Yeast can be the difficult one - it's much easier here as it's relatively easy to scavenge dregs from pub casks or there's a few container-conditioned options in supermarkets. And there's Brewlab when they're in the mood to send out slopes to homebrewers. Although S-04 and Nottingham (or eg Windsor/S-33 followed by Notty for flocculation) are widely used by modern microbreweries, they don't have the character of the house strains of the regional breweries. I've not used it but it seems that Imperial A09 Pub has a lot of the marmalade character of Fuller's yeast (which WLP002 and 1968 lack). Lots of people like 1469, I'll put in a shout for WLP041 Pacific Ale - despite its name it's a British yeast that gives the easy drinkability which British beer should have.
Aim for a quick, healthy fermentation and don't allow the yeast clean up after itself - see
this thread on HBT.
British beers are all about balance - water, malt, hops, yeast, CO2 and alcohol all in balance with no one aspect dominating. If you have problems, then you need to adjust the process or recipe in the direction that restores balance.