Chemical attacks are relatively easy to detect and can be spotted quickly. A number of systems are in place (or are being put in place) to alert us to the presence of toxic chemicals. Having said that, with such a huge amount of potential targets to choose from, there's every chance that a chemical attack will have done most of its damage by the time any detection system kicks in.
Chemical agents would probably be delivered in gas form (with a crop duster or aerosols) or in liquid form (with a crop duster, aerosols or contamination of water supplies). Another feasible scenario is that a common chemical agent like phosgene would be released in the air by blowing up a tanker or chemical plant. If you live close to a chemical plant, you should already be aware of procedures in case of an emergency.
The situation is different when it comes to biological attack. There are few detection systems that can pick up a biological attack (although some advances are being made on this front). It is most likely that it will take several days (depending on the incubation period and the concentration of the agent) before we recognize that we've been attacked.
The most likely scenario is that we would become aware of a biological attack when doctors begin to notice an increase in patients exhibiting the same symptoms. It is hoped that our doctors will be better trained in the ways of biological agents so that they can be on the lookout for suspicious symptoms.
The exception to this is anthrax which is not contagious. In the event of an anthrax attack, the scenario would probably be closer to the description of a chemical attack (above).
If an attack occurs outdoors - whether it be chemical or biological - the agents will travel with the wind. It will not take very long for the agents to be dispersed in the air and diluted to the point where they present no further danger. The amount of time it takes depends on a number of factors like wind-speed, humidity, the concentration of the agent, temperature, and so on.
If an attack occurs indoors (in a large building or on the subway), then the agent will be carried through the ventilation systems. In the case of a chemical attack, once the gas has dispersed the situation is over. However, in the case of a biological attack it may be just the beginning.
In most cases the biological attack itself will pass by unnoticed and will only show up, as I've said, when people start displaying symptoms. By then, depending on the incubation period, the disease may be considerably spread - even to cities and regions well away from the initial attack.
Attacks using the biological agent anthrax or most of the chemical agents is more likely to take place indoors where sufficiently dangerous concentrations of the poison can be more easily reached.