Usually drive down(using the M6 Toll which is a lot easier around £10.00 Depending on day and time of day).
You can also take the train,may I suggest that you take a direct service as you don't want to be carrying your luggage from train to train! (be aware that there is not a lot of space for cases on these new Virgin trains).
Flying down to Southampton is also an option,which we have used.Its very quick and less hassle than the train and it can be cheaper(or it was for us). The only problem is you are going on anything longer than a short 2 or 3 nighter the luggage excess can be expensive.
If you do fly down ,there are plenty of ways to get to Southampton City Centre-
Trains are very frequent(to get a ticket there is a machine on the platform that accepts credit and debit cards also buses are very good(but with cases!!),Taxis are not waiting outside (unless you book them prior to getting there,through the link above)waiting time if you book one when you come from the aircraft can be up to 45mins.
Just click on any of the links above to take you to the relevant site.