South England Itinerary

First, I'd suggest more time in London. Second, with limited time, I wouldn't head all the way out to Cornwall or Devonshire - too much distance to cover in a short time. Bath, or nearby, would give excellent access to the countryside as well as the high quality sites in the area (Roman Baths, Stonehenge, Salisbury Cathedral, Wells, Glastonbury, etc.) I'd do something like this:

Day 1 - London (Stay in London)
Day 2 - London (Stay in London)
Day 3 - London (Stay in London)
Day 4 - London (Stay in London)
Day 5 - London to Bath (Stay in Bath or nearby)
(hit Windsor castle, Avebury or Stonehenge, and Somerset/Wiltshire countryside on the way)
Day 6 - Bath (Stay in Bath or nearby)
Day 7 - Bath, may be visit Wells/Glastonbury (Stay in Bath or nearby)
Day 8 - Cotswolds (Stay in Cotswolds)
Day 9 - Cotswolds - may be visit Stratford or Blenheim Palace (Stay in Cotswolds)
Day 10 - back to London and onto Paris

My sister and I stayed in Bath last June and in addition to doing the lovely town of Bath and the must see Roman Baths we also rented a car and drove to the lovely small towns of Glastonbury and Wells, did a day at Stonehenge/Old Sarum/Avebury/Silbury Hill. You could also do Salisbury itself and its' Cathedral. I haven't been to the Cheddar Gorge but would like to see it, and with a little research you could find some nearby ruined castles as well. If you decide to base from Bath I would recommend 3-4 days there with the corresponding nights and the rest of your time in London. We spent an entire week in London and there was more to see that we didn't get to so I would spend 4-5 days there and maybe leave the Cotswold's for another time.

The Cotswolds and the southern counties (Cornwall, Dorset, Somerset, Devon, W Sussex, E Sussex) all sound lovely

What about Hampshire! ; )

You could pick any one of those counties to spend three days in and barely scratch the surface. I could give you an itinerary that would cover the south coast of Hampshire and West Sussex that could easily exceed three days but that would exclude the other areas that interest you.

Look at the National Trust UK web site for some great itinerary ideas.

I agree with the first comment that you may want to focus on Bath - Cotswolds - London. However, instead of trying to see Stonehenge and all the other sights in the region on the way from London (which are really not all on the way), I would use your rental car to do a full day's loop from Bath. I would also skip Glastonbury unless you're very interested in crystals and tarot.
So:
Day 1: arrive London, train to Bath

Day 2: recover from jet lag, see Bath sights

Day 3: rent car in Bath, drive a loop to Stonehenge and Avebury

Day 4: drive to the Cotswolds village of your choice

Day 5: enjoy Cotswolds (drive around, walk, whatever)

Day 6: ditto (and perhaps do laundry etc.)

Day 7: drive to a convenient town to drop off your car (Oxford?), or else drop it off at Heathrow, use public transportation into London and check into a London hotel

Days 8 & 9: London

Do make sure you reserve a car with an automatic transmission so you'll only have to concentrate on driving on the left.

Day 3: rent car in Bath, drive a loop to Stonehenge and Avebury

But the OP hasn't expressed any interest in visiting neolithic sites. Whilst Stonehenge is a major tourist attraction I would only recommend visiting if you actually have any interest in it. I see no point when time is so limited on visiting somewhere just because everyone else does. The OP stated "we also want to explore small villages, gardens, fishing villages, foot paths, pubs, manor houses, etc. in the countryside", with so much of that throughout the country it is a difficult enough choice as it is.

With all the people on this forum headed to the Cotswolds I am surprised the UK doesn’t tip sideways into the Atlantic Ocean. There is a whole other half of the country that seems to be largely ignored - the East. There are lovely villages in Essex, Norfolk and Suffolk as well as Kent, East and West Sussex, and yes, Hampshire!

A lovely trip from London is to Brocklehurst in the New Forest. The forest is a remnant of the woodlands that covered the UK centuries ago. There are wild ponies who wander freely through the woods and keep the village greens trimmed. Pubs, etc are all over the place. Visit Jane Austen country around Chawton. Head to the coast, Chichester with its cathedral and the nearby Roman Palace of Fishbourne. Bosham Harbour a small village just outside Chichester with a Saxon church. Legand has it that Bosham is where King Canute tried to turn back the tide.....Beaulieu House is nearby with a ruined Abby in its grounds and a wonderful collection of antique cars.

If the Cotswolds is a must it is relatively easy to get there from Hampshire. Have a look at Cirencester as a possible base. A lovely town, with a very interesting and long history. The museum is a gem with incredible Roman mosaics. Bibury is a classic Cotswold village and is close by as is the Roman Villa at Chedworth. Oxford would be a good stop on the way back to London and as the university term probably wouldn’t have yet started you would probably be able to visit the lovely college grounds.

Also have a look at places to the east/northeast of London. Cambridge is another University town with lovely colleges and grounds. Ely with its cathedral, Norwich with a Castle and Cathedral. The old Huguenot wool towns in Suffolk such as Lavenham.

"A lovely trip from London is to Brocklehurst in the New Forest. "
Brocklehurst?? Did you mean Brockenhurst??

I was going to suggest Chichester as well! Perhaps visit Goodwood on the way down. Take in the beautiful estuary, Hayling Island and then go from Portsmouth over to the Isle of White. Then take the return ferry to Lymington and finish up in the New Forest.

Isle of Wight!!!

Interesting choice regarding Hayling Island, it's not somewhere I would suggest unless you're keen on wind or kite surfing. Other than that it's just residential areas with a few holiday parks and a crappy amusement park thrown in. The only reason I go there is either when I'm policing it or to visit a friend who lives there.

Too early in the morning JC!

Ok, yeah, the Chichester side has probably more charm than Hayling Island. Add in Uppark House maybe?
OP as you're open to not doing the usual tourist thing I'd strongly recommend not taking in the Cotswolds and enjoying all these lesser explored areas.

Definitely the Chichester side of the harbour although the coastal walk from Emsworth to Langstone is very picturesque with some good pubs and restaurants along the way. If feeling rigorous enough you could continue walking the coastal route either to Portsea Island or to Portchester and visit the castle.

Thank you for all of the fantastic responses. You've certainly given us a lot to think about. I will add that we aren't that interested in the stone monoliths or ruins, as our time is limited. The suggestions in the southeast (Essex, Norfolk, Suffolk, Kent, East and West Sussex, Hampshire) sound lovely, and we're going to look into those. A couple questions: Is there a fast rail that goes to a smaller city in that general area that would allow us to get out of London on public transport and rent a car in that city, and explore from there? We'd rather not have to rent a car in London. Also, is it too ambitious to think that we could explore that southeast area for a couple days and then drive to the Cotswolds for another couple days? Again, all of your feedback has been tremendously helpful!

That's a BIG area to explore. The UK might be similar in size to California but there is a lot crammed into it.

My advice.....forget the Cotswolds. I know everyone goes there, I know it features on pretty much every American's itinerary but that's because it's a self fulfilling recommendation. People see it recommended so feel that they should go and you end up with a snowball effect with often people visiting when they don't really understand why they are other than it's a "must see". Yes it's an attractive place but no more than many other similar villages in the UK, most without the hordes of tourists thronging the streets and driving up prices.

You could easily fill three days in Norfolk or Suffolk or East Sussex, West Sussex or Hampshire. Three days is no time at all. To put it into perspective we recently spent a day at Mottisfont https://www.nationaltrust.org.uk/mottisfont, a house and gardens near Romsey, Hampshire. It's a 50 minute drive from our house and it consumed almost an entire day. That's one visit in one county in one day! Of course you can cram more in if you wanted to but there's a limit and isn't the whole point of travelling abroad to actually experience and enjoy the location rather than rush through ticking things off?

Most people tend to rent a Car from Gatwick or Heathrow, both of which are way outside Central london and straight onto the motorway. If that doesn't work out for you however, an option might be to get a train from London Waterloo to Southampton airport parkway. A direct train that takes just over an hour. The train station is across the road from a choice of car rental outlets. If you choose this option, book your tickets as soon in advance as possible as an advance single is as little as £9.10, where as turn up on the day and an off peak fare is £41.

Yes, definitely consider reducing the amount of places you're planning to visit, especially if you're planning on driving, UK roads can get busy! Hampshire is a beautiful county and not overly touristy if you're after a more authentic experience.
Winchester, the original capital of England is well worth the trip and potentially a good base to explore from - it's just an hour away on the train from London Waterloo. http://www.visitwinchester.co.uk/city-centre
From there you can travel to the beautiful New Forest (actually one of the most ancient in the UK) and see the wild ponies, plus Stonehenge isn't too far away either. http://www.newforestcentre.org.uk/
Good luck!

You mention a small city in Suffolk as a possibility.

Well, one hidden and not much known (outside the UK anyway) is Bury St Edmunds, Suffolk. Just over an hour by car up the M11 from London, or a similar time from London Liverpool St by train.

Beautiful medium sized town (it does have a Cathedral, but isn't a city because there is no Bishop).

Lovely Abbey Gardens in grounds of ruined abbey (had second highest tower in Christendom before the dissolution).

Medieval grid of streets and town centre. Traditional market on Wednesday and Saturday every week.

In the unlikely event you don't like it - it's just 26 miles east of Cambridge, so pop along there instead.

But, it's a lovely place, and you'll love it!

https://youtu.be/6sLBIBp9638