Hi I used Southampton a couple of times over the summer from Guernsey.
You can request handeling direct from the airport (not signature) who charged me approx £45 for a PA32 with an instrument approach. The only snag is if you want to stay parked for any length of time. The time limit is 2hours. Over two hours you have to use Signature and the cost is significantly higher.
You will likely be directed to the Northern Apron and parked by marshaller. They have a car that will meet you and take you to the security gate. From there its a left to the main arrival hall which has a costa, toilets etc.
Return through the same gate (through the porta-cabin) the car will meet you. They take credit card payment in the car! It's pretty straightforward for a busyish airport.
Call airside operations on Tel: 023 8062 7102 they were very helpful. They have limited parking so usually wanted to know pretty precise times.
Have a good flight, remember the hi viz !