You know what? You're absolutely right. Donald Trump is not your president. He's not my president either. In fact, he's not anybody's president. At least, he's not anybody's president until around Noon on Friday, January 20, 2017. That is when he will become your president (assuming you live in the United States) and my president. He will be our president whether we like it or not.
Sure, you can try to delegitimize him like upset conservatives did in 2008 after President Obama was elected. The question is whether you can do so without deserving whatever accusations of hypocrisy might come your way. If you had no problem with Republicans seeking to delegitimize President Obama, then you should be in the clear. De-legitimize away! On the other hand, if you complained when Republicans did this to President Obama, then you may want to rethink things a bit, assuming you are interested in avoiding hypocrisy.
The good news, of course, is that recognizing the reality that Trump will be your president after January 20 in no way prevents you from criticizing him or working against whichever of his policies you do not like. The even better news is that you can criticize him, work to oppose his policies, and even be part of the anti-Trump resistance movement without necessarily trying to delegitimize his presidency. Opposing Trump's agenda through strategic resistance is not the same thing as seeking to delegitimize him or the office he is about to take.
As for the ridiculous attacks on Trump's physical appearance and his wife's appearance that have been so popular on social media, I'm honestly not sure what their purpose serves. If this is all you've got, you've got nothing. Remember, many of you complained bitterly when Republicans did this with the Obamas. Are you really going to go down the "But it's okay when we do it" road? That particular road is a dead-end. Repeatedly sharing these memes on your social media accounts does not seem to be helping.
My advice here is simple: let your passion energize your activism, but let reason guide the approaches you take. If you really want to resist Trump, do so effectively. Keep the focus on the issues that matter. Direct your energy toward criticizing Trump's many bad ideas and the policies he will seek to implement with which you disagree. Let us know why you think his ideas are bad. Let us know how you think his policies will be harmful. And if possible, suggest some alternatives and let us know how we can help. Connect with your elected representatives and let them know that you expect them to oppose Trump's worst policies. When they do so, express your appreciation. When they fail to do so, make it clear that this will negatively impact their chances at re-election.
One last suggestion. Over the next four years, you will encounter political conservatives and libertarians, some of whom may have voted for Trump. Some of them will share your displeasure with at least some of Trump's ideas or policies. Recognize that they are potential allies in your efforts to resist Trump. Instead of going out of your way to alienate or shame them, connect with them around those areas where you both oppose Trump.