Getting to the ground
The ground is a 20-25 minute walk from the station, and around the same from most of the city centre pubs mentioned further down. The route from the station is apparently signposted, but I wouldn't expect anyone to be walking straight to the ground!
If you're driving down then parking is a bit tricky. Don't even think about the car parks around the ground unless you don't mind being stuck for an age after the game. You used to be able to park in the streets just south of the Western, which was handy for the pub and for a quick getaway, but that area is all residents parking now. The nearest street parking to the ground is now the Compton Road area, just off the A5460. However, there's a railway and a canal between there and the ground! The shortest walking route is a mile, and it's shown on the pub map (bottom). Alternatives are the Lothair Road area, off the A426, or Clarendon Park, off the A5199. All three street parking suggestions are indicated on the pub map.

If you're in the car then the options aren't great. Most of the pubs near the ground are home fans only or are to be avoided. An exception is the Counting House, which is a family pub. If you park around Compton Road then you could call in at Westcotes Constitutional Club, which is more or less on the way to the ground. It lets away fans in free, and is family friendly. Otherwise the obvious place to head for is the excellent Western, which is a reasonable 10-12 minute walk from the ground. It's very friendly, and they have a good range of hand-pulled beers. Be warned though that it doesn't open until 12. It'll get very busy towards kick-off, so you could consider walking another 5 minutes or so to the highly rated Black Horse or the West End Brewery . If you park around Lothair Road then the best option is the Cricketers (allow 20 minutes to get to the ground). The New Road Inn (pretty poor, 15 minutes to the ground) or the aforementioned Counting House are the choices for anyone parking in Clarendon Park.
For train travellers the options are numerous. A good bet is the Barley Mow, a very traditional Everards pub with pool table. Just past the Barley Mow is the Last Plantagenet, the nearest Spoons to the station. My personal recommendation for a city centre pub though is the King's Head, a Black Country Ales outlet and another place that doesn't open until 12. It's small and cosy, the service is fast even when it's busy, and it's relatively cheap. There's an excellent range of beers - three of their own plus around nine guest! A gem a bit closer (15 minutes' walk) to the ground is the Sir Robert Peel (12 o'clock opening again!). There are quite a few other good pubs in and around the centre, including the Blue Boar, Rutland and Derby and Bowling Green (Wetherspoons clone).

Pub map
CAMRA Leicester pub guide (slightly out of date, but otherwise good)
Football Ground Guide
Visiting supporters guide (from official Leicester site)