Southampton:
unilink buses conveniently connect Highfield Campus and Glenn Eyre Halls with the City Centre, Southampton Airport and the trainstation. Here is a network map and here are the time tables.
For a local taxi ride call 023 8066 6666 (Radio Taxis).
Winchester:
You can go to Winchester either by bus or by train from Southampton Central Station or Airport Parkway Station (both train station can be reached with unilink buses).