Take NH 1 from Delhi upto Ambala. At Ambala, turn north towards Shimla on the NH 22. At Zirakpur, take the Bypass to avoid Chandigarh and Panchkula.
Solan is on the way to Shimla on NH22... not very much to see though.
To stay at Shimla you will have to think of a suitable place to park the car?
You can opt for the HPTDC hotels-
Hotel Peterhoff or Hotel Holiday Home (Cart Road).
HPTDC properties always have car parks within the premise. Contact them for more details, you can book online too.http://hptdc.nic.in/hotel.htm
http://hptdc.nic.in/custsw/viewuser8.asp (for online booking)
At Solan, there is HPTDC Tourist Bunglow, which is a budget hotel. Instead of staying at Shimla, which is very crowded , you stay at Chail Hotel Himneel. From Chail you can visit Shimla. And Chail is higher and prettier than Shimla... anyday! Kufri , Fagu, Naldehra, Tattapani (Hot springs on Sutlej banks), Mashobra, Narkanda, Sarahan can be visited from Chail / Shimla. Do visit Sarahan, although it is a 5 hrs drive from Chail, the best place in Shimla District, i should tell you.
Have fun!