We're also burning oil in cars of course. Fortunately, electric cars have become incredibly popular thanks to Tesla. Once electricity generation is clean, electric cars are perfect for taking advantage of that.
Planes and ships are going to be hard. On top of that, many ships burn incredibly dirty oil, and the fuel for planes and ships is untaxed, which, in a world where all other forms of energy are taxed, make them effectively subsidised. International treaties are necessary to make ship fuel cleaner, and extra efficiency may be encourages by taxing fuel. For short distances, planes may be replaced by high speed trains, but for long distances, I don't think we'll be able to get rid of them.
Fortunately, there are other things we can do: trap CO2 from the atmosphere by planing trees, encouraging other plant growth, and possibly even stimulating algae growth in the oceans because that's where the real large scale capacity for this is. There might be other ways too.
At least in part, this can be accomplished simply through better policies by governments, maybe international agreements. But to encourage businesses and people to burn less carbon, I think a growing tax on the emission of CO2. I don't see the cap-and-trade working very well, and even if it does, it merely limits it; it doesn't bring it back to zero. A carbon tax can do that. I think the carbon tax should start low, because you don't want to kill the economy with crippling taxes, you just want to make it attractive to switch to cleaner energy. The tax should then be used to either invest in cleaner technology, or to directly remove CO2 for the atmosphere. You could even pay companies to do that for you, creating a new industry that cleans up the pollution from those businesses that are unable to make the switch for whatever reason. At first, the tax won't be enough to clean up all CO2, but eventually it definitely should be. And once you're there, it doesn't really matter if some people or companies still pollute, as long as they're also paying to undo that damage again.