First of all, as Han Solo once said to Luke Skywalker, "Don't get cocky." I realize that asking any politician, even one whose experience is mostly outside of politics, to reign in his arrogance is a tall order, but it must be done. You will soon be the most powerful man in the world (which already scares the [bleep] out of the left), and will have responsibilities more grievous than most people would ever want. If memory serves me correctly, George Bush the Elder once said that there really is nothing that truly prepares you for the presidency. For that reason alone, a little bit of humility will go a long way.
While I urge humility, this does not mean that I would try to appease the Democrats in Congress by allowing any of Barack Obama's political appointees to remain at their positions. If the experience of George Bush the Younger is any indication, this sort of thing will not do you any good. Those who hate your guts will continue to hate your guts no matter how well you treat them, or how much you "reach out" to the other side of the aisle. This does not necessarily mean, however, that you should never appoint a Democrat. For example, there is an informal tradition, going back to Bill Clinton, of presidents appointing a member of the other party to a Cabinet position. But whomever you appoint to whatever position, regardless of party, should be someone whom you can trust to help implement your policies. And even though most of them are your fellow Republicans, veterans of the administration of the above-mentioned Bush the Younger should face at least as much scrutiny as anyone else.
I understand that you have already decided to designate the Muslim Brotherhood as a terrorist organization (H/T Red Fox Blogger). They have also been declared a terrorist group by Egypt, Saudi Arabia and the United Kingdom. I would also suggest rooting out and getting rid of any members of the MB in the federal government, who might be holdovers from the Obama administration. Don't get me wrong. I'm not saying that you should keep Muslims out of yours. The Constitution bans religious tests for public office. But membership in a terrorist organization, regardless of the inspiration for the terrorism, should automatically disqualify anyone.
You may get some criticism if you "walk back" any of your campaign promises, but I think you could scale down your proposal to stop immigration from Muslim countries until we get our vetting system fixed. There are quite a few Muslim countries where terrorism and terrorists are very uncommon. You probably won't be endangering too many Americans if you allow us to take in people from places like Bangla Desh, Uzbekistan or Bosnia. But on the other hand, as events in places such as Boston and San Bernardino have shown us, our vetting system is seriously messed up, which you were right to notice. As for immigration in general, simply enforcing our existing laws and allowing the Border Patrol and ICE to do their jobs will go a long way. I don't know if there is any realistic way to make Mexico pay for it, but by all means, please build that wall.
You will soon be inheriting military operations in both Afghanistan and Iraq/Syria. During the last eight years, I have often heard complaints about the rules of engagement in those two areas. I would thus urge a serious review of these rules, with input from military officers and NCO's, and where needed, appropriate modifications.
By now, you've probably heard about how America's military veterans have been mistreated, even by the agency that's supposed to make their lives easier after they leave active duty. According to the United State Justice Foundation, veterans have been declared incompetent to handle their own financial affairs by the Veterans Administration, and forced to prove otherwise. If they can't, they are no longer permitted to own or use firearms. Moreover, these cases have not involved medical personnel or judges, but only bureaucratic employees of the VA. If a veteran then appeals his case, the VA suspends his or her benefits until the appeal is decided. You can read this article at the USJF website for more details, and find other similar articles listed here. (H/T Southern Sense) I must concede that I don't know to what extent these things are occurring, but I hope that the people you appoint to oversee the VA will make sure that the Second Amendment and due process rights of veterans are respected, and that any effort to legally deprive any veteran of his or her right to keep and bear arms is undertaken only with competent and qualified medical and judicial personnel, and without the veteran losing benefits while appealing a decision. As you might appreciate, the people who took up arms to serve our country deserve those rights as much as anyone else.
Two other things I would recommend reviewing are the Patriot Act and the NDAA. Every once in a while, I seem to hear that either or both of these laws allegedly can be used to deprive American citizens of their Constitutional rights. For example, the federal government can (allegedly, again) declare any American to be a "terrorist", and as such, detain him indefinitely without charge, without any particular due process. Maybe I'm being paranoid here, but if there are parts of these laws that need fixing in order to protect the civil rights of American citizens, you should ask Congress to make the appropriate changes.