First, an easy one: no, the president doesn't have the power to "end Common Core" as has been his rhetoric. In fact, under ESSA, there is explicit language barring the Secretary of Education from setting standards. The states set standards, and the only thing the federal government can do is attempt to persuade them in particular directions (usually through funding).
Can Trump, as he has said, get rid of the federal department of education? It's unlikely. He would need Congressional coordination to do so (as that's essentially a budgetary decision), and there's a great deal of funding that flows to states through the US DoE. Even those who have long complained about federal education oversight would be hardpressed to tell the areas they represent why they have lost the state millions of dollars. For a perspective on just how much comes from US DoE, see here for what those losses could look like. How that funding comes through to states could well be rearranged--essentially, as EdWeek points out, implementing some of the amendments to ESSA--to block grants or something similar. That is absolutely something to watch for, as, for example, lumping Title I in with other programs could result in fewer dollars getting near poor kids (and likewise through the titles).
(editing to add) Part of his 100 day plan is the following:
School Choice And Education Opportunity Act. Redirects education dollars to give parents the right to send their kid to the public, private, charter, magnet, religious or home school of their choice. Ends common core, brings education supervision to local communities. It expands vocational and technical education, and make 2 and 4-year college more affordable.SO that's federal vouchers. It's not clear where the money on that is coming from, if it's Constitutional (state by state) and how that would work; would states have a choice? Vouchers, for the record, have already been found unconstitutional in Massachusetts.
He also might well cut the Department itself, and that is where my biggest concern is right now:
...they’ve said there’s no need to keep the department’s Office for Civil Rights, which oversees Title IX enforcement and has become increasingly active over the last eight years as the spotlight on campus sexual assault increased. But eliminating OCR would be “absolutely devastating to survivors and educational access in this country,” said Alyssa Peterson, a policy coordinator at Know Your IX, a group that advocates on behalf of sexual assault victims.This is also the office that send out guidance around inequalities around student discipline, has ensured equal access to education regardless of legal status, and has spoke to districts regarding Title IX and transgender student access. The link above says that states then will have to step up their own enforcement; the problem is that some states will not. This absolutely leaves students vulnerable.
More as I have it.
UPDATE to add this link to the Hechinger Report, which can't find much more, but has some Massachusetts mentions.