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The answer very much depends on where you are. Kissing on the cheek or "air kissing" toward the cheeks is reserved for close friends or acquaintances. It is never correct when first introduced to someone. It is traditionally done to both cheeks by the cultures (such as in France) for whom it is acceptable.
In some cultures it is forbidden to touch a person of the opposite sex, so shaking hands is not allowed, and would be a major faux pas.. Up until lately , a lady did not shake hands at all. She just said "how do you do?" and perhaps inclined her head politely.
In modern culture in the west, a woman is as likely as a man to offer her hand when introduced to either a man or another woman. It is best to let the woman make the first move. If she offers her hand, by all means shake it. If she doesn't, smile and speak the polite greeting that is expected.
Please let me know if you have questions.