The reflected light is not completely polarised. It's certainly true that $s$ polarised light (i.e. light polarised in the plane of the surface) is more strongly reflected, but except at the Brewsters angle $p$ polarised light is reflected as well.
The plane of polarisation of $s$ and $p$ polarised light is not changed by reflection/transmission, however a mixture of polarisations will be changed because the $s$ to $p$ ratio of the reflected and transmitted light will not be the same as the incident light.
The reflection and transmission of the light can be calculated using the Fresnel equations. The link I've given is to the Wikipedia article Fresnel equations and this is an excellent starting point if you want a thorough explanation of the subject. To go much farther you'll need a good book on Optics. My favourite is "Optics" by Hecht and Zajac, though I think that can be hard to find these days.