Yes, the weather will be horrible. I'll have about two weeks and I'll be spending most of them in Dharamsala and other areas nearby. But it's my first, and probably only, trip to India so I'd like to see the Taj Mahal. So I'm thinking of spending a few days in Delhi and Agra. I'm guessing that I would be better off arranging a driver but would like to know for sure. Any other suggestions on dealing with the weather would be appreciated.

FWIW I used to live in an area where it got above 100 degrees F for about 100 days/year. I now live in an area where it routinely gets to the upper 90s for 3-4 months. But both areas have dry heat and I live there and don't go out and do touristy stuff during the middle of the day.

Thanks in advance for your help.

Drink plenty of water. North Indian heat is severe and dry. You can dehydrate quickly.
Dharamshala will be better than Delhi and Agra, obviously.
If you are in Agra, do try and also take in Fatehpur Sikri.

Late may/early June are probably the worst time to be in India. It can be above 110 degrees F , and it feels a lot hotter due to about 100% humidity. Nothing like desert temps. At least in Northern India, most hotels and taxis have AC, but definitely take an umbrella for the sun.

Dharamsala in up the mountain, so a bit cooler. However, it is a major faming area at the bottom and they burn the crops in the time period when you'll be there. It can really obscure the views, and it does get quite smokey, in Mcleod Ganj.
all your hotels can arrange local or long distance drivers. You can also take the train to Delhi and Agra. Another option is to fly from Gaggal airport (in Kangra at the base of the mountain) to Delhi, then use drivers once there to sightsee or go to Agra.

It will be scorching. yes drink plenty of water and wear light colored clothing and get a good hat. I don't mean a baseball style cap which holds the heat in. Splurge a little and get a Tilley hat. They will last a life time and provide far better protection. Just google them.

Columbia and Eddie Bauer and perhaps others have clothing that is impregnated with sun block. I have some of their shirts and wore them to Kuala Lumpur last year. They do help.

Good luck and as a general rule buy travel insurance.

Northern India could like fire at this moment, you have suffer dehydration at this time.

Delhi is 107 today.