I want to read Plato's The Republic, Aristotle's politics, and Machiavelli's The Prince. Is there an order in which I should read them to better understand the development of political philosophy?
I recommend reading them in the (chronological) order in which you have listed them. This for two reasons linked to your rider, 'to better understand the development of political philosophy'.
Aristotle critiques Plato's Republic in Politics, II, and elsewhere. You can make better sense of Aristotle's criticisms and more accurately assess their relevance and fairness to Plato if you first know the Republic. So first Plato, then Aristotle.
Machiavelli's The Prince marks a particular point in the separation of politics from ethics. Stage (1) : For Plato, the just person and the just city are indelibly associated in the Republic. Plato doesn't at least in the Republic take much interest in the working of politics apart from ethics. Even his account of the decline of the ideal polis in Rep. VIII & IX presents a logical model of political decay, not a political prediction. Stage (2) In the Politics, Aristotle does begin to show an understanding of and an interest in the the practice of real-world politics; the book takes normative politics into account but not at the expense of the empirical study of politics. The autonomy of politics as a sphere of life having its own tendencies and dangers is coming into view. Machiavelli's Prince does not reveal his political preferences; those are to found in his Discourses where he favours a form of republican freedom, a community of self-governing citizens. Stage (3) The Prince does, however, focus on politics largely independently of ethics. Its aim is to enumerate and specify the actually effective qualities needed by a ruler, especially a single ruler, in order to maintain power and control the citizenry. There is no guarantee that these qualities will be used with ethical correctness.
There are actually 2 answers to this question; so you should choose which answer best suits you:
- Begin with Plato's Republic, followed by Aristotle's Politics and then read Machiavelli's, The Prince. This is the classic, History of Political Philosophy curriculum, which begins with Plato-(who was essentially, the Father of Political Theory), followed by his student, Aristotle and then Machiavelli-(who was probably the Father of Modern Political Theory).
- Begin with Plato's Republic, followed by Machiavelli's, The Prince and then read Aristotle's Politics. This type of order is not historically based, but rather, a collection of Political Philosophies that primarily examine the nature of singularly held Power-(i.e. Plato's Philosopher King and Machiavelli's, The Prince) versus Representative or Citizens' based power-(i.e. Aristotle's "Politea"). After reading these contrasting works, you will see how Machiavelli may have been somewhat supportive of Plato's Philosopher King, though he probably would have disdained Aristotle's Representative "Politea" or Citizen's Government.
Either way, you should begin your Political Philosophy education with its most landmark work....The Republic by Plato.