Southampton Airport uses one terminal common to all flights. Domestic and international departures and arrivals are handled at one location.
The terminal is very small and it is impossible to get lost here. Security checks tend to be efficient and queues of around 10 minutes form at most in the morning departure peak. As a general rule, just arrive at the airport 60 minutes before your flight.
There is never a passport check on departure, even for international flights. Conversely, if you are arriving in Southampton from abroad, you will go through passport control.
Transfer between flights
Southampton may not be a typical connecting airport, but for travel to the Channel Islands in particular, it's sometimes worth combining a transfer on your own here. In fact, flights to Jersey, Guernsey or Alderney are quite frequent from Southampton.
The terminal does not have a direct transfer option, so you have to go outside and go through security again on arrival.