Europcar have an office next to the historic dockyard which, as someone has already mentioned, is well worth a visit. The easiest way to get there is via a short taxi ride (it is walkable from the ferry terminal however if you're carrying luggage you might want to opt for a taxi). From the Europcar location it is a very easy, short drive to the M275 motorway which will take you on to the M27, M3, A34 and eventually the M4 towards the Cotswolds.
Another option is Enterprise car rental who will collect you from the ferry port and take you to the rental location. The drive from there is also relatively easy as you take the Eastern Road which is a dual carriageway leading all the way to the M27.
Unfortunately there are not really many rural areas in the vicinity that have car rental. There is an Enterprise near me in Havant which is located on an industrial estate and is a very easy drive onto the A27/M27 in less than 5 minutes. A taxi there from the ferry port would probably cost around £15. This would be the best option if you wanted to avoid driving out of a city and will get you on to a motorway very quickly where you can take your time adjusting to driving on the left. Unfortunately there isn't an easier way to get to the Cotswolds from Portsmouth, driving is by far the easiest and quickest, it took us just over 1.5 hours last Sunday. The route we took (although we were coming back from Derby via the Cotswolds) was M27, join the M3 after passing Southampton airport, come off at the slip road for the A34. Continue along the A34 and join the M4. This will take you past Swindon and a GPS (very advisable to have one) will direct you to the main towns. You will probably want to hit Bourton on the Water first and then follow the road up through Stow on the Wold and Moreton-in-Marsh.