Car and driver in Edinburgh

If you look for 'Private Hire Taxi Edinburgh' in a search engine a few should pop up. Hopefully Unclegus can recommend some of the companies.

never use taxi companies, but I am sure many of the Private hire companies will sort something out for you, my late mother used BlueBird cars a lot. http://www.bluebirdprivatehireedinburgh.co.uk/
where is the cemetery might be able to advise on how to get there by [public transport.

Yup, told you he was good for a recommend!

You might also try UBER. I know they have have drivers in Edinburgh that would be able to do this for you.

The cemetery is the "old" cemetery in Newbattle. We thought maybe we could hire a car and driver to take us there, wait a half hour and then take us to Craigmiller Castle and leave us to explore it. We would take the bus back to our hotel from there. We've done that before. I thought public transport might get too complicated and time consuming but would be willing to give it a try if there weren't too many transfers and the timing works out. There are four of us which is another reason we thought a private car might not be that out of line expense-wise. I appreciate any help or suggestions.

Newbattle is a good bit outside.the city might take 30 or 40 minutes to get there from the city centre.
I actually live quite near Craigmiller castle and pass there every day to an from work.

https://www.rome2rio.com/s/Edinburgh/Newbattle-Cemetery
you could take the train from Waverley to Eskbank on the new borders rail line and you have a short walk from there, train will only take about 15 minutes, then take a bus 29 or 49 from Eskbank (Tesco Store) to the New Royal Infirmary (that's where I work) and there is a path from there right up to Craigmiller Castle, less than 10 minute walk.

Thanks Unclegus. We have taken your path from the infirmary up the hill to Craigmiller Castle several times but not for about seven years so it is good to know you can still do that. We are staying at a hotel on Lauriston St. in Edinburgh and I thought that from there a car and driver might not be too expensive to do what I wanted as I believe the Newbattle cemetery and Craigmiller castle are in the same direction and not terribly far apart. I did contact one taxi company who quoted me a price of 55 pounds but said it could be more depending on congestion. I really wanted a flat rate for two hours. Does 55 pounds sound fair? I will try to call Bluebird as we get closer to our departure date. I didn't find an email address for them.

£55 sounds pretty reasonable to me.
If you are using the buses to the Royal infirmary they have changed to the other side of the main hospital.The new Sick Children's hospital is being built where the old bus stops and car park was, getting near completion now.