How we cite our quotes: (Paragraph.Line)
Quote #1
Since its very outset, it has been found necessary to conduct our Government by means of political parties. That system would not have survived from generation to generation if it had not been fundamentally sound and provided the best instrumentalities for the most complete expression of the popular will. It is not necessary to claim that it has always worked perfectly. It is enough to know that nothing better has been devised (13.2-4).
For an opposing viewpoint, visit Shmoop's handy guide to political parties. The framers of the Constitution thought that parties would be the absolute worst thing that could happen, because then narrow factions would promote their own interests at the expense of the common good. Now, however, we know that that never, ever happened.
Quote #2
When the country has bestowed its confidence upon a party by making it a majority in the Congress, it has a right to expect such unity of action as will make the party majority an effective instrument of government. This Administration has come into power with a very clear and definite mandate from the people (14.1-2).
In this section of the speech, Coolidge talks about how political parties have a responsibility to uphold their party platforms. Basically, the idea is "you were elected because you said you'd do this, so you'd better do it or else you're a liar liar pants on fire."
Coolidge's landslide did imply a mandate, but all presidents claim it even if they just barely squeaked out a win. Or lost the popular vote.
Quote #3
We are not without our problems, but our most important problem is not to secure new advantages but to maintain those which we already possess. Our system of government made up of three separate and independent departments, our divided sovereignty composed of Nation and State, the matchless wisdom that is enshrined in our Constitution, all these need constant effort and tireless vigilance for their protection and support (20.1-2).
Can you name those three separate departments? 35% of Americans can't even name one (source). Coolidge gives the nation a mini civics review here; maybe Facebook should slip some of this info into everyone's news feed. Anyway, Coolidge probably brings this up to emphasize the limits on executive power and the respect for local governance.