Actively seek correct information on climate change and your role in causing it, and take action accordingly.
This answer may seem annoyingly vague. The problem is that no one action is best for everyone.
Most of the global warming since 1950 is very likely to have been caused by the sudden spike in greenhouse gases: mostly carbon dioxide from burning fossil fuels, but also methane from cattle and other sources. By now carbon dioxide concentrations have shot up from 290 to 390 parts per million. If we continue 'business as usual' we may easily reach 1000 parts per million or more by the end of the 21st century. This will give the Earth's climate a push whose rapidity and magnitude has no known precedent.
To slow climate change, it is therefore crucial to reduce carbon emissions. However, carbon emissions are distributed very unevenly among the Earth's population. For example, in 2007 the average human put 4.4 tons of carbon dioxide into the air through their use of fossil fuels and cement - but citizens of the EU put out 11.8 tons, and for the US citizens the figure is even higher: 18.9 tons!
So, what's best to do depends a lot on who you are. If you're a politician, maybe you can help eliminate some of the $300 billion worldwide yearly subsidies to the fossil fuel industry subsidies that waste taxpayer money and encourage the burning of carbon. If you're an average American, taking one fewer round-trip flight from New York to Los Angeles can save 3/4 of a ton of carbon dioxide. If you're a subsistence farmer in Africa, you can turn some agricultural waste into charcoal and bury it: biochar is a promising low-tech way to sequester carbon. In many places, but not all, you can try to kick out politicians who fail to take climate change seriously. The best ways to do this vary from place to place.
Thus, broad generalizations are difficult. But for everyone, the key step is to start actively seeking correct information on climate change and your role in causing it, and then take action accordingly.
Some places to start include the Union of Concerned Scientists' webpage:
Despite the somewhat silly answer posted above, the user is right - becoming a vegetarian, or simply eating less meat, would be a huge contribution if a large number of people did it. Right now, the meat production industry is a key contributor to greenhouse gas emissions. Methane (CH4) is a strong greenhouse gas, stronger than CO2, meaning that it traps heat more effectively than CO2, so smaller amounts of methane may cause more warming than CO2.
Eating less meat would be significant. The average American eats meat with 2-3 meals per day - much more than what is recommended for a daily diet. We really don't need that much protein in a day, and what we get from meat we can get from other sources.
Another thing to do would be to drive less! Walk, bike, and take public transportation whenever you can, driving only when necessary.
Because I am not the azimuth project nor wish them to represent me, I will suggest that the one best thing everyone can do to slow down climate change --- supposing climate change to be a bad thing everyone can do something to slow down, that is --- is to learn to recognize, prize, and practise virtue.
Of particular importance will be those facets of virtue known as self-control or self-restraint --- the government of reason over the passions, that is --- and proper obedience to authority.
One useful exercise in acquiring the practise of virtue in respect of self-restraint is the deliberate sacrifice of unnecessary comforts. This may be aided by practising a habit of contentment --- of finding happiness in simple necessity. Particularly, one should regularly make a habit of eating what food one knows to be sufficient, whether you still feel peckish afterwards or not; of moderation and prudence in lovemaking; of wrapping up in the cool when heating is not needed, or dressing lightly and keeping to shade when air-conditioning is not needed; of spending only what one must of the money one actually has; of speaking politely in polite company. Spinoffs abound.
Now, when I say this is something everyone can and should do (I hope that's obvious anyways, but still... ) I really do mean everyone. I mean the poor farmers and the poor bankers, and the rich bankers and rich farmers, and carpenters and biologists and private soldiers and generals and bureaucrats and governors and sons and daughters and parents and hermits and ... And by authority I mean not only ligitimate offices, but authentic true wisdom, consciences well formed and safe engineering practices. Heavens forefend, there may come a time when, for the good of the planet, it is really found prudent that a billion people have their roofs white-washed, so to increase the ground albedo and cool the earth. (this seems like a fair example, as it is simply coordinated and reasonably reversible); in such an event, it were obviously advantageous if a billion or so people can be found who will assent to the call, and that not a million hinder the project in painting all their roofs dark green!
But it's equally important that any decision to whitewash a billion roofs not arise from some influence of the lime-and-chalk industry, but from a genuine need to make certain regions of the earth more reflective. And so, anyone calling the shots ought to be really informed by such advisors as ought to know, and the advisors ought really to be delivering true knowledge and neither wishful thinking nor self-interested deceit.