Looking at the entire Declaration of Independence, what are ten of the most important concepts that we might expect to be included in the Constitution that would be written in 1787?

In some of the most famous lines in the Declaration of Independence, we find important concepts that would later be incorporated in the Constitution:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.—That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed.

First is the idea, also incorporated in the Constitution, that (white) men are all equal under the law. The Constitution will incorporate the right to fair treatment and expressly prohibit the creation of an aristocracy.
Second, liberty (at least for whites) is treated as an important concept, one that is enshrined in the Constitution.
Third is the idea that governments are man-made and are structured from the bottom up—deriving their legitimacy from the consent of the governed. This was a radical idea at a time when most governments were understood as divine in origin, with a monarch being appointed by God. The opposing concept of "consent of the governed" is enshrined in the Constitution.
Fourth, the declaration forcefully rejects the tyrannies it says are imposed by George III. These include forcing people to quarter troops in their homes at their own expense. This was abolished by the new US government. Fifth, the Constitution sets up taxation so that elected representatives establish the tax laws, which gives people a say through voting in how they will be taxed. Sixth, jury trials (which the declaration says George III has denied the Americans) are allowed in the new nation. Seventh, George III is accused of dissolving or not allowing legislative bodies to meet, another act that will be prohibited in the new republic.
Eighth, governments are instituted for the good of the people as a whole, not just the few on top. Ninth, people's rights are "inalienable"—they come from a divine source and cannot be taken away. Finally, implicit in this document is the idea that republicanism is a superior form of government to monarchy because it safeguards the rights of the people in the form of laws that cannot be transgressed.

1). Representative government. One of the Americans' main grievances was that they were taxed without having political representation. In drawing up the Constitution, they would seek to remedy this deficiency.
2). Equality. All men are created equal and therefore have the same inalienable rights as everyone else.
3). Limited government. The long list of grievances set out in the Declaration could only be remedied by the establishment of a system of limited government. The colonists didn't want British tyranny to be replaced by an American tyranny. Government was much stronger under the 1787 Constitution than it had been under the Articles of Confederation, but it was still relatively limited all the same.
4). Republicanism. The colonists had had enough of being King George III's subjects. They felt that the time was right to have a republican system of government with an elected head of state. This was duly provided for by the Constitution.
5). Separation of powers. To ensure that no one branch of government dominated the others or accrued too much power to itself, the powers of the government would be divided between a judicial branch (the Supreme Court), a legislative branch (Congress), and an executive branch (the presidency).
6). Government exists to defend our inalienable rights. If it doesn't do that, then the people—"We, the People"—have the right to change it. The Constitution would establish a number of mechanisms for changing the system of government such as the provision under Article V, which sets out the requirements for holding a constitutional convention.
7). No taxation without consent. Most of the American colonists recognized the importance of taxation, seeing it as a necessary evil, but what they wouldn't accept was the imposition of taxes without their consent. Under the Constitution, no new taxes would be imposed without the explicit authorization of the nation's political representatives.
8). The government is a government of laws, not men. The Declaration roundly condemns what it sees as British tyranny, of its riding roughshod over local laws and customs. The new government must be based on the rule of law as expressed through the judicial power. This means establishing a system of courts.
9). The military power should be subordinate to civilian authorities. The colonists hated standing armies, seeing them as potential instruments of tyranny in the hands of the British, but what they hated even more was that the military was effectively independent and superior to the civil power. That would all change in the Constitution, which would give Congress the power to raise and maintain armed forces.
10). The American colonists's rights are natural rights—rights that are theirs by virtue of being human. These rights have been given by God, and as such, they are inalienable.