Staying in place is good advice in most circumstances.
As a CRUDE guide, being inside an average house reduces the radiation dose to one tenth of that received in the open.
A deep concrete basement reduces the level to about one hundredth of that received in the open.
A basic but purpose built fallout shelter might achieve a thousand times less radiation than outside.
A sophisticated shelter with air filtration plant, de contamination showers and the like would be about ten times better i.e. reducing the dose to about one ten thousandth of received in the open.
Standards of construction vary a great deal, so the above is an approximation.
So ten hours fleeing and out in the open has perhaps given you the same dose as one hundred hours sitting at home, or a thousand hours in a deep basement.
The best course of action also depends on the nature of the incident.
The effects of a small nuke as might be used by terrorists, are fairly local, and moving a mile away MIGHT help a lot.
In the case of large detonation, the effects are very widespread and moving a mile would be unlikely to help much and could give you a significant dose whilst so doing.
Fleeing might be the best option in the very early stages before a few million other people have the same idea.
For example, when I worked in London I had two different plans according to whether I was actually at work or within a hundred yards, or if I was out in the open.
If AT WORK or very near, shelter in place, I had a deep concrete basement stocked with supplies..
If out in the open, and I believed that a nuclear incident had occurred, I would IMEDIATLY hail a taxi, not stopping to twit nor tweet, nor to make phone calls, or take photographs.
I would go to the nearest main line rail station, in the direction away from the incident, and get on the FIRST TRAIN to anywhere else.
I would not stop to buy a ticket or make any enquiries.
A dust mask, or better a proper gas mask gives considerable protection against inhaling radioactive particles, as has already been said.
Remember that the used filters may be HIGHLY RADIOACTIVE and should be disposed of with care.
Water or other drink should be fine if from sealed containers that are wiped clean before opening.
Water from other sources should be filtered before use, remember that the used filters may be HIGHLY RADIOACTIVE.
Boiling or chlorination is of no use whatsoever against fallout, but may be prudent against other hazards.
A home emergency kit or bug out kit should include a Geiger counter. EDC of a Geiger counter is arguably a bit OTT unless you have some specific reason to be concerned.