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The time you would have to wait all depends on the credit card. From my experience (and from reading other posts on this topic, per the search function, as the previous poster recommends) these are the time frames you must wait before being able to apply for a new card.

American Express (Delta, Starwood, etc.): Can't churn these cards. Only one signup bonus allowed.

Citibank (AA): 8 weeks

Chase (CO, BA, UA): 6 weeks

US Bank (NW): Can't churn these cards. Only one signup bonus allowed.

Bank of America (US): ??

Juniper Bank (US, Midwest, etc.): Can't churn these cards. Some people have posted that they have received multiple signup bonuses but if you read the terms and conditions they state that you can only receive a bonus once, and if you do, they reserve the right to charge you for the cost of the miles or revoke them.

Most card issuers don't make you wait a certain amount of time before applying for a new account after you cancel an existing one, but they do limit how often they will consider offering you credit (i.e. every 6 weeks, 8 weeks, etc.). In theory (and based on my experience: in practice) you can cancel a card and re-apply for very next day, or even later that same day.

If you don't already have too many cards with a given issuer (usually the limit is 3-4 (or more) depending on credit history, income, etc.) you can usually have two or more ACTIVE cards of a certain type (i.e. two Chase United MasterCards) from a given bank.