How do I determine the demand curve of a product through a survey? I'm a student and I want to determine the demand for a product (and consequently determine what will happen if the price of the product increases). I don't have access to any businesses that can provide the data for me (price and demand at that price?

How do I estimate the demand curve or function by surveying people? What I currently have in mind is to ask people what amount of the product they'll demand at a given price and extrapolate this data for a city or a region. 

Additionally, what'd be a good sample size for such a survey?
If it is a small project, you may benefit from reading this

At most, you can estimate price elasticity of demand using your survey data to make predictions/forecasts. For this you need to estimate a demand function as @EconJohn suggested, but you need to include additional variables such as income, price of a substitute or complement etc. to hold them constant (i.e. ceteris paribus). Note that the demand function does not allow you to draw the downward sloping demand curve (like the one in textbooks). It is very hard, in practice, to draw actual demand curves using survey data. Even when you have the most complete data, without including the suply function in your estimation, you face so called identification problem. In short, econometric estimation of demand curve does not allow you to draw a downward sloping curve of the type you see in textbooks.

The link above has a good explanation about this point, and it can be used as a guide for your project.