be'ezrat - literaly means 'with the help of', and you can use it outside religious context too:
העיוור הולך ברחוב בעזרת כלבו.
The blind man walks on the street with the help of his dog.
'baruch' means blessed, and it inherently carries religious association, but you can use it on whoever you want, not only god, although it's less common.
And as Shira said, hodot le'ariel, bizchut ariel, are the proper ways to say 'thanks to...'