If you're going down by train then the best way to get from the city centre to the ground is to catch a train from New Street to Witton, which is only a 5 minute walk from the away entrance. In many cases a through ticket to Witton is the same fare as to New Street. There's a more frequent service to Aston than to Witton, but Aston's a 15 minute walk from the away section. If for any reason the trains aren't running then you can catch the number 7 bus instead. The stop in the city centre is on Colmore Row (stop AE), which is very convenient for the pubs on Bennetts Hill. Coming back you should get on at the stop on Witton Road just past the station. There is actually a stop nearer to the away section, but there's likely to be more of a queue there. These stops are marked on the pub maps below. The bus runs every few minutes on a Saturday and every 12 minutes on a Sunday.
If driving down then there's street parking around the ground, although not as much as there used to be now that residents only parking has been introduced in the around the Witton roundabout. If you park anywhere near the ground you'll probably find that it takes you absolutely ages to get away after the game. You might therefore want to consider parking somewhere around Lichfield Road (ie near Aston station). Even though it's a bit of a hike from the ground, you should get home much earlier!
Visiting Villa Park is like going back to the dark ages. The only pub near the ground that you can be sure of getting in is the Witton Arms. This is known to many people as the Cap and Gown, but it's apparently reverted to its original name. They might have changed the name, but it's still crap! You have to pay 2 quid to get in, then the beer and the service are rubbish. The nearest alternative is the Yew Tree, which should be ok. In any case, the best pub anywhere near the ground is the award winning Oakham Ales pub the Barton Arms. It is though a 20 minute walk away. As well as an excellent range of beer, they also do Thai food. A further option is the three pubs next to Perry Barr greyhound stadium. One of them, The Arthur Robertson, is a Wetherspoons. Again, you're talking about best part of a 20 minute walk from the ground.
For anyone going down by train the best bet is to stay in the city centre and get a train or taxi to the ground late on. There are quite a few decent pubs in the city centre nowadays. The best of the bunch is the Wellington, roughly half-way between New Street and Snow Street stations. This place is an absolute gem, having no less than 16 hand-pulled beers on at any one time, as well as a good selection of foreign beers. The only slight drawback with the pub is that it doesn't do food; however, they provide plates and cutlery for anyone who wants to consume their own food on the premises. To get to the Wellington from New Street, follow the signs for the Stephenson Street exit, and then follow the route shown on the city centre pub map below. For Wetherspoons fans there's the Briar Rose, just down the hill from the Wellington.
City centre pub map
CAMRA Birmingham pub guide