can you post some photos of the area in question, especially the existing dwelling that the roof will connect to? what are the dimensions of what you want to build?
a few questions:
what building code are you under?
will you need a building permit to construction the patio cover as well as the foundation?
are you looking for a concrete foundation wall or piers for the columns for the roof?
will plans/drawings be needed to submit for a building permit?
what is your minimum depth of footing/frost protection requirement?
what is the ground snow load for your area?
where are you located?
if your building code is based upon the 2009 International Residential Code a patio cover supported only by columns this not prescriptive code. Prescriptive code requires walls to support roofs not only for gravity loads, but lateral loads such as seismic and wind. you may be required therefore to have this designed by a professional engineer or architect that will certify the design and construction are in compliance with the building code. By prescriptive I mean if you do the steps exactly as described in the code then you meet the code. if it is not described within the code then you must have someone certify that it complies with the code.
now you could be in an area that neither has a building code, or requires building permits. this doesn't mean for some strange reason your area is beyond the effects of the physics of building construction, it just means the local town/city/county doesn't care what you do. if you care about the safety of your family or the investment of your money and time then you should build it properly.
footings are typically sized based upon the loads they must support and the soil bearing capacity of your soil.
some answers to the above questions, and a few photos will provide you with more detailed information.