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. 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 and Dry Dock (both nothing special, 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. Straight over the road from the station is the Hind. The reviews are at best mixed, although it seems to have improved since an enthusiastic new manager took over recently. A much better 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. Plus 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, a few of which are shown on the pub map below.
CAMRA Leicester pub guide (slightly out of date, but otherwise good)
Football Ground Guide
Visiting supporters guide (from official Leicester site)