We have just spent a week in Hyderabad, we choose this destination because of the direct flights from Europe, so going there for a short time seemed OK
It is not a city for people coming to India for the first or even third visit; we have been to India several times so we like these out of the way places.
We really liked the city, very helpful and kind people, only saw 6 tourists during the week.
The prices for hotels are quite high, as many places in India but we stayed in 2 hotels, that we found reasonable priced.
Because of the Diwali causing a lot of traffic jams, we divided our time to different parts of the city.

Woodbridge Hotel in Lakdi-ka-pool, paid 2240 Rps for double aircon including breakfast.
The room was OK, a bit of traffic noise, staff quite helpful, breakfast very Indian.

The next hotel we consider a real find: Minerva Grand www.minervagrand.com
has only been open 6 month, Japanese style, red and black and a lot of open space, we had a suite, 50-60 m3, paid US$ 125, no traffic noise at all. The staff has been very well trained.
The hotel is in the Paradise area of Sekunderabad, 10 min. walk to Paradise restaurant, lively street with a lot of local shopping.

What really made our trip, was our taxi driver, he just happened to be driving, the first time we called for a taxi. He drew for us every day, a very pleasant young guy, knew the city in and out, walked with us through several markets, so we did not get lost.
If you go to Hyderabad, we can highly recommend him. His name is Susheel,
e-mail: [email protected] (it is neo_ andrisonn) and cell 99483 33592

We spent most of our time driving / walking through streets and markets, people watching, enjoying the decorations for Diwali. It is one of the most pleasant and easy going cities we have been to in India.

Colkonda Fort was a nice trip out of the city, we did not stay for the light show

Ramoji Film city was in our opinion a vast of almost a whole day

We also a bit of shopping:

Pearl jewellery: somewhere on the internet I had found recommendation of this shop:
Princess Pearls, Peshwar Plaza, South Abids. It is a whole sale / retail shop, did not buy any expensive stuff, but quite a lot of earrings 200 – 450 Rps each to use as presents. To us it seemed very good prices compared to other shops. If you do go there say hello from DK for us.

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We are looking at a RTW ticket that would let us go to Bombay, Delhi, Chennai or Hyderabad. My choice was Hyderabad, but your comment about this not being a good place for your first trip to India has me wondering about that choice. What would you recommend? We will be coming from South Africa.

My suggestion would be Delhi or Mumbai.
How long time do you plan to stay in India, are you only going to stay in the city or will you have time for the countryside?
I think Delhi would give you a better idea as to what India is all about than Mumbai.

Probably about five days. We are retired, in our late 60s and healthy, but I don't think we want to spend a lot of time in India. We will also be going to Bangkok, Saigon, and Hong Kong as well as several places in Europe and Capetown.

You have not mentioned what time of the year you are going.
I still think Delhi would be the best for you, have a car and a driver/guide showing you the sights, and then I know everybody else would suggest you to go Agra to see Taj Mahal, but instead I would spend 1 or 2 nights at Neemrana Fort. It is only 120 km outside Delhi.
Take a look at www.neemranahotels.com.
Staying at these old palaces and forts has been the highlights of our trips to India.
Not many other places in the world, where you can get this experience.
I do not know your budget for hotels, but the cost is quite high in India