For my taste go first to Huntercombe, only a few miles south-east of Oxford. Historic Park stuff.
Tadmarton Heath is a lovely course outside Banbury with many quite treacherous gorse-clad holes. It's not long but it is very cunning and the drainage is superb, making for good winter or poor-weather golf. It's not overgrown with trees, either. It's one of my favourites.
Southfield is the University course - Colt I think - and there are some quite challenging holes on it but it is girt about by housing and an old military barracks. I haven't played it for some years so I cannot comment on its current condition. When last I played there (some time in the 80s) it was not in good nick at all. It gave courtesy of the course to many US Servicemen during the Second World War who were stationed nearby and when I was an undergraduate in the late 60s there were several members who could remember the exploits of some of those GIs who had been professional or good amateur players. The 400-yard 3rd, for instance, was for us a spoon shot downhill to avoid running on into a stream, and then a horribly demanding shot to the green, long enough that we needed a spoon again, yet with that we could not make the steep climb up the hill to the green. Even during the war these chaps habitually flew the stream with their drives (240+ yards?) and could make the height to the green with a short iron. The 7th is fun - a drive from a high gun-platform tee over a stream and many trees and the chance of a strong man driving the green (320 yards) but lots of trouble for those who just fail. The 374-yard 11th is a fine 2-shotter reminiscent of the 1st at St Andrews and the par-3 12th is a wicked hole on high ground with terrible misfortune awaiting the slightest pull and no room to chicken out on the right. As we played it, the 13th then involved a cape-hole drive over (out-of-bounds) allotment gardens (complete with greenhouses) but when I was back there in the 80s these had been abandoned and it was then a drive over a wilderness of brambles, elderberries and nettles. The effect on a topped shot was the same, but nettles are far less satisfying to hit with a foozle than greenhouses! I once had an eagle-2 on this hole. I love it!
North Oxford is a gentle parkland course but nothing out of the ordinary.
Frilford Heath is worth playing (though it is now rather expensive) for the Red and Green Courses which are good heathland tracks - not all that old, but well in character. However, 2-ball play is confined to one of the courses and 4-ball to the other and these are alternated day-by-day, so you can't play both courses on the same day unless you are a four who are prepared to split up or play foursomes. Avoid the Blue Course - it is a modern thing of limited appeal.
The Rees Jones Oxfordshire is expensive and a great blot on the landscape.
There's an affordable public course at Waterstock which is well designed (Steel, I think).
The rest of this part of the Cotswolds are not particularly well served.
Otherwise, outside Oxford, have a look at Goring and Streatley or Henley - both beautifully situated courses with lots of interesting holes in seriously hilly country.
You're not far from Newbury and Crookham which is a lovely old course, dating from the 1920s and originally designed by John H. Turner, the Oxford University Professional. The club later amalgamated with an older one, founded in the 1870s, so you'll see 1873 on the club's crest. It utilises the lie of the land beautifully and there are any number of subtle holes. It's right next to Greenham Common where there was a huge USAF Cruise Missile base which attracted years of anti-war demonstrations but the base is now closed and the course is a charming rural retreat.
If you venture further west in the Cotswolds try Broadway - great views and some challenging holes - one of the driest courses in the country.
If you go north east, see if you can get on Northamptonshire County a part-Colt course of considerable variety, lovely setting and a touch of class.
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