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.