If you are not a Saudi local it is okay not to wear a veil or cover your hair. You do find a lot of expats in Jeddah and Riyadh with western clothes (i.e. without Abayah, Burkha or Head scarf).
Respecting their local culture it would still be a good idea to for women to wear conservative clothing when in the Kingdom (i.e. no shorts and jacket or long sleeved shirt and no tight t-shirts). Wearing a head scarf if you are comfortable with it would help you blend better with the locals.
It is also not common for women to shake hands with men.
I lived in Saudi (Jeddah) for six months in the early 90s. So my knowledge is a bit dated although I dont think anything has changed much. Women have to certainly cover their head in Jeddah, Riyadh etc. The only exception is made in Al Khobar / Dammam where white expats can get away without that. To my knowledge (actually I have personally witnessed a pretty negative scene), Indian women have to wear the full burkha; otherwise, they might come to the notice of the religious police called Muttawa, which is not a good idea.
Women have a pretty strict regime there. They cannot go out alone; they have to necessarily be accompanied by a male from the family. They also cannot drive.
Having said all this, I found Saudi to be an awesomely pretty country with wonderful landscapes.
