to the ground
To get to Stamford Bridge you catch the Piccadilly line from Kings Cross to Earls Court, from where it's the District line to Fulham Broadway. It should take just under half an hour. It's then a 5 or 6 minute walk to the ground. Alternatively, you could just walk from Earls Court (20 minutes), particularly if you opt to drink in one of the pubs around there.
Many of the pubs around the ground are very partisan, if not to be avoided altogether (eg Imperial and Jam Tree). There's a Lloyds No 1 (the Oyster Rooms) in the shopping centre above Fulham Broadway. It gets decent reviews, but I don't know if they allow away fans in. A lot of away fans tend to stick to the pubs around Earls Court tube, before making their way to the ground late on. There are plenty to choose from. but they're all pretty upmarket, with fancy/expensive food. Three of the nearest are the Courtfield, the Blackbird, and Prince of Teck. A much cheaper option is the Goose on North End Road. It's about a 15 minute from both Earls Court (or half that from West Brompton tube) and the ground, but it's a Wetherspoons clone, which is reflected in the beer and food prices. A final suggestion is the White Horse at Parsons Green, which gets really good reviews. You stay on the tube for one stop after Fulham Broadway.
Kings Cross pub map
Football Ground Guide