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 the obvious place to head for is the excellent Western. It's a reasonable 10-12 minute walk from the ground, which is far enough away for it to be very friendly. They offer 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. Another option is the aforementioned Huntsman. 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 (medocre reviews, 15 minutes to ground) or the 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 recently refurbished Hind. Much better 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. 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! A couple of gems a bit closer (15 minutes' walk) to the ground are the Swan and Rushes, which has bar billiards, and the Sir Robert Peel, but both don't open until 12. There are quite a few other good pubs in and around the centre - see the CAMRA guide below.
CAMRA Leicester pub guide
Internet Ground Guide