A good plan would be to fly into Heathrow.
Follow the signs within the terminal to the Central Bus Station at Heathrow airport.
Buy a bus ticket to Oxford at the ticket window. Comfortable bus to Oxford.
Stay in Oxford a couple of nights.
This helps you get over jet lag and explore.
Someone mentioned Blenheim Palace as a place to see. You can get there by bus or taxi from Oxford.
Blenheim is impressive. We were on the Rick Steves Best of England in 14 Days tour in April, and the tour went there. But it was not my favorite place on the tour. I don't think I would go way out of my way to see it if I went again to Oxford; but that's just my personal preference. Decide for yourself, of course, whether it's worth it for you.
Take the train to Stratford-Upon-Avon from Oxford. (Warwick Castle is a stop along this route. Hop off; see the castle.)
Stay two to three nights in Stratford-Upon-Avon
I disagree with those who say it's too tacky to visit. There's still a lot to see from Shakespeare's day. The theatre there is great. Walk around town after the tourist crowds have gone for the day. Really nice. Lots of good restaurants and pubs.
From Stratford-Upon-Avon, you could take a tour of the Cotswolds with this company: Go Cotswolds.
My husband and I did this in spring of 2016 and enjoyed the Go Cotswolds tour very much.
It is an all day tour, leaves from Stratford-Upon-Avon.
Tom will pick you up at your hotel and return you in the evening.
Transportation is in a new Mercedes mini van.
https://www.gocotswolds.co.uk/
Villages on the tour include Chipping Campden, Stow-On-The-Wold, Bibury, and Bourton-On-The-Water, to name just a few of the stops. You stop in these villages, are given a map and time to walk around there.
https://www.gocotswolds.co.uk/tour-details/
About the Cotswolds. There are many beautiful villages there, but there are similarities to all of them. A sampling is enough to have an enjoyable day out, and that is what you will get on the mini-van tour we took. We enjoyed not having the hassle of a rental car there. Don't have to worry about returning it with scratches on it, or getting lost.
If you were staying in the Cotswolds for a long time (2 or 3 days, or a week or longer) I'd say, yes, rent a car. But for a short one day visit, I'd say no, a car is not necessary.
My husband and I have been to England many times, but do not enjoy driving when we are there. We enjoy riding the trains and buses. You can get to many places in England just by riding the train. Most train stations have taxis there waiting for you. This is helpful if you need to get to your hotel with a lot of luggage.
You can take a train from Stratford-Upon-Avon to Bath or to London.
If you only have 10 days, you will have used up a chunk of time by the time you get on the train in Stratford-Upon-Avon and head to London. I might save Bath for next trip. (But I do love Bath) There's a lot to see in London, and you will soon run short of time there.