If you're driving down then parking is a bit of a nightmare, as is getting away after the game. Street parking on the north of the city centre is non-existent - it's all residents only. There's a Park and Ride specifically for away fans near junction 8 of the M27, but the reports on it are not good (like the first bus isn't until 2 hours before kick-off!). Possibly the best option is to park in the 'Ocean Village' area (where the Cove pub is - see pub map). It'll cost you a few quid, but after the game you can be on the M27 (via the A3024 and A334) in next to no time. If you end up parking in the city centre then you probably want to avoid the A33 after the game. I'd suggest giving the A335 a try instead, or possibly Hill Lane and then Winchester Road across to the A33.

St Mary's is within walking distance of the city centre which means that there are plenty of pub options even though most of them aren't particularly close to the ground. Nearest is the King Alfred, although this tends to be home fans only for 'big' games, and that's if it's open at all. It did close down, but I've heard that it now opens on match days. The Chapel Arms is less than a 10 minute walk south of the ground and welcomes away fans. Not too far from the ground in the Northam area are Browns, the Cove and the Prince of Wales. There are quite a few pubs in the Bevois Valley area, including the Guide Dog, which is generally reckoned to be the best real ale pub in Southampton. Not far away are The Alexandra and The Rockstone, which also get good reviews. The pubs around here are a 20-25 minute walk from the ground, but this should mean that they don't get too busy. If you're on the train then you could do a lot worse than the Giddy Bridge which is an above average Wetherspoons outlet. There are also a load of pubs on Oxford Street.
All the pubs mentioned, plus a few more, are shown on the pub map.

Pub map
Football Ground Guide
The Ugly Inside (good unofficial Southampton site)