It's difficult to narrow this down to merely two things, but if I had to, I'd say:
1) Government works for you, not vice versa. You need to hold it accountable to the contract (Constitution). You need to criticize them when they do wrong, and praise them when they do right. You need to agitate so that government continually improves.
2) You have responsibilities as well as rights. You need to be informed, teach your kids of their rights and responsibilities, show up for jury duty, pay taxes, be a citizen of the government in order for it to function. But as I said in #1, not a blind, compliant citizen.
Two important things to remember about our government: I don't know that there are two most important things to remember about our government. Maybe there are and I just don't know about them.
Very important to me is the fact that the right to vote is not restricted to people whose life-experience enables them to understand how government impacts their lives (which is too bad). Rather, nearly everyone 18 years old or older can vote. Most people have not the foggiest idea of how government affects their lives; most people lay far more importance on the Super Bowl than on the national election. Therefore, most people can be manipulated fairly easily as regards how they vote in national elections. They vote according to a few seconds of information that they get via their television sets. To get on TV costs lots of money. Lots of money flows to approved candidates from big business. Therefore, elections are between two candidates that are preapproved by big business. Those not preapproved by big business do not get enough money to get on TV, therefore few voters ever hear of them, and never hear of them as often as they hear of the two preapproved candidates.
Another thing that I think is important is that the president is not elected by a simple national majority, but by electoral votes apportioned amongst all the states (which is a good thing). If election of president were by a simple national majority, there would be enough people in just a few of the most urbanized states, who thought alike, to elect the president by themselves. Therefore the president would not give any concern to the most of the states which did not figure in the outcome of the election. Those states would become mere imperial territories of the few states that elected the president.
There will be many important elements to come out of this question. I would say that one lasting legacy of our government would be its emphasis on individual rights (In the term "ours," I am going with American Government.) The legacy of American Government has been one that has placed a large primacy on the notion of individual freedoms and singular liberties. This has been an element that has been present since before the nation's origin and something that is still heavily emphasized today. Application of knowledge of rights based concepts is critically important. I would say that another lasting legacy of our government is the idea the democracy can be messy and one should not be afraid of having to examine this mess. That is to say that there have been many issues that have been very complex to arise in society and government and this nuanced approach should not be feared or resisted. Rather, it must be embraced with a sense of openness and transparency in order to resolve such intricacy. Issues that have connected to slavery, racism, discrimination, women's reproductive rights, rights of the accused, and national security vs. individual freedoms have all been debated, discussed, and even to this day are not resolved. Their complexity, intricacy, and "messiness" is what helps to define a strong government, and one that has been used to define ours.
Goodness... two most important things?
I would say one of these is that our government is set up in a way that is meant to make it hard for it to do things. For example, the President and the Congress are elected separately and the President has no control over the Congress. You could use that to understand why things seem not to get done in our system.
Maybe the second thing is that the people have a great deal of power on issues that they can understand, but less on ones that are difficult and technical. Perhaps you can apply that when people start saying the big businesses and the interest groups run the country. You can understand that the people run the country on issues that they care about and understand. For example, no company or interest group could force through a law either banning or allowing all abortions -- the people wouldn't allow it.