What is the process of deciding domestic policies and their priorities?
If you are talking about the president and his (someday her) process of deciding domestic priorities, every president goes about this in somewhat of a different way. This is decided partly based on the president's own personality and partly based on circumstances.
For example, Pres. Obama's first priority when he came into office had to be the economy. Circumstances forced him to put the economy before all else because the country was in the middle of a crisis.
When circumstances do not dicatate what the president must do, he often goes based on what he thinks is most important. This can be based on his ideology. For example, if a Republican wins in 2012, they might focus on cutting the size of government because that is what their ideology tells them they should do.