How do I start an essay about how the Women's Rights Movement and the Gay Rights Movement strengthened the Civil Rights Movement?

I think you need to begin such an essay by establishing the fact that all three of these movements derive from a single, historical impulse, one enshrined in the Declaration of Independence, whose second paragraph states:

"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...

I think you need to begin such an essay by establishing the fact that all three of these movements derive from a single, historical impulse, one enshrined in the Declaration of Independence, whose second paragraph states:

"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." 

The principle that all men have inalienable rights and that the United States was formed in part to protect these sovereign rights, serves as the bedrock on which all civil rights are based. You must explain that The Civil Rights Movement was a movement to extend the full panoply of rights already guaranteed by the constitution to African Americans, regardless of where they lived.

You can also argue that the 14th Amendment (Equal Rights Amendment) and the 19th Amendment (which gave women the vote) are simply part of the larger Civil Rights movement, to extend the rights that white men had enjoyed to women and minorities. The Gay Rights Movement is just another facet of the extension of Civil Rights to minority groups. In the case of the Gay Rights Movement, the people in question are those who face discrimination as a result of their sexual orientation. 

The extension of equal protection to all classes of citizens (including women, ethnic minorities and the LGBT community) strengthened and continues to strengthen the Civil Rights Movement because the whole notion of "inalienable rights" falls apart if those rights are denied to any group of people.

In other words, if our government can deny rights to any group of people, then those rights are not inalienable. Remember, our Constitution says that the government does not and cannot confer rights or privileges to any group, or deny them. The government can only protect those rights from being infringed upon, because everyone is born with those rights. Therefore, any government that selectively protects the rights of some citizens while denying those rights to others cannot claim that its citizens have inalienable rights. 

Clearly, a government that can deny rights to a class of individuals has "alienated" or "separated" those people from their basic civil rights. That kind application of the law is what constitutional lawyers deem “arbitrary and capricious,” completely at odds with any notion of equality, and therefore, a threat to our democracy.