Pete Escobar published an article in the Hong Kong press when Trump took office that is a must read for anyone that wants to understand his administration’s approach to foreign policy, particularly US-Russia relations. Read the article here. Virtually everything Trump has done with regard to foreign policy fits with the paradigm described in this article.
The source of the Trump administration’s foreign policy ideas appears to be the Center for Strategic and International Studies. Trump’s foreign policy approach seems to be a move away from the neocon paradigm that has dominated Republican foreign policy approaches since the Reagan period, and a return to the Nixon/Kissinger approach of the Rockefeller Republicans. This is consistent with Trump’s New York background. His administration is moving back toward the Nixon/Kissinger approach and away from the neocon Republican or liberal internationalist (neocon lite) Democratic approaches. He’s been more of a neocon on domestic policy than on international relations, though in a very pragmatic way.
Trump’s economics seem to be the standard right-wing Keynesian approach that has dominated the Republicans since the Reagan period, although on trade he has departed from that a bit (although Reagan did so as well on a selective basis).
My guess is that a lot of folks among the power elite now regard having let the neocons run foreign policy as a mistake, and as having created too many liabilities and are now pushing them to the side. Notice even the neocon mouthpieces at FOX, who have been reluctant Trumpians, are getting upset about the Putin summit. The only exceptions have been Tucker Carlson, the token paleocon on FOX, and Hannity, who only cares about his ratings, and the fact that most of his fans are diehard Trumpians.
Thus far, Trump seems to be pursuing Nixonian detente with Russia for the purpose of reducing the potential for nuclear conflict, expanding the petroleum trade with Russia, containing jihadi terrorism in Central Asia, and wooing Russia away from an alliance with China.