Getting to and from the ground
Once at Lime Street the easiest way to get to Goodison (about two and a half miles away) is to jump in a taxi. There's loads of them about and they're cheap - if there's more than three of you in a cab (and they're licensed to take five) it'll work out cheaper than a bus. Alternatively, you could catch a Merseyrail train from Liverpool Central or Moorfields to Kirkdale, 15 minutes' walk from the ground (see Goodison pub map below). There are eight trains an hour, and the journey time is 8-9 minutes from Liverpool Central, or 6-7 minutes from Moorfields. The advantage of this is that it doesn't cost you a penny as a ticket to Lime Street is valid through to Kirkdale. After the game you could either go back to Kirkdale, or there are special buses back to the city centre from Walton Lane. If you're lucky the bus will get you back to Lime Street in time for the 1722, but you can't bank on it.
If you're driving over, try to get there reasonably early and park up near Anfield and you'll be well placed for getting away after the game.
As far as pubs are concerned, it's horses for courses. There's a load around the ground and in particular along Walton Road and County Road, and the only one I'm aware of as being totally no-go for away fans is the Royal Oak. It goes without saying that the nearer you are to the ground the busier they're likely to be. There's a Wetherspoons (the Thomas Frost) on Walton Road, although it does apparently get chocka. Just down the road is
Bradleys Wine Bar (not actually a wine bar, just a pub!), which was totally refurbished in 2009 and gets good write-ups. If you're driving over and park somewhere near Anfield you might want to drink in the one of the pubs around there. The Arkles is an old favourite.

If you get the train over then you may prefer to drink in the centre until late on, in which case you're spoilt for choice. We have in recent years gone to the Richard John Blackler just down from the station. However, it's notorious for bad service so I would give it a miss. My recommendation is the Ship and Mitre, which has a vast selection of beers, from both the UK and abroad. From here it's a five minute walk to Moorfields station. Another option is the Globe. This a traditional cosy little pub, with great staff and friendly customers, not to mention a good range of beer. And the Globe is literally two minutes from Liverpool Central station! Other places you might want to try are Doctor Duncans and the Poste House, both of which have a good range of beers and get decent reviews.

Goodison pub map
City centre pub map
Internet Ground Guide
Official Everton site
Liverpool pub guide (very comprehensive and fairly up to date)