I have a 12 gallon aquarium (salt) and want to upgrade to a 24 gallon.
Most people who start that small quickly want to upgrade too. 
I have crabs, shrimp, liverock and 2 clowns in the current tank.
K - that's a good stock, 12gal is pretty small for 2 clowns, but 24 is def better - though 55gal's for a pair would be better, LOL. But since you have had them for a while, and I assume they are doing well, we won't get ahead of ourselves. I am sure they will really appreciate the 24gal. Also, which type of clowns do you have?
Is it okay to just set up the 24 gallon, fill 1/2 with their old water and 1/2 with fresh and move them over to the 24? Of course, I am moving everything, including rock.
Hmm, well if it were me, I would move over everything. Using only 6 or less gallons of matured water for a 24gal would mean that 18 of those gallons would be pure tap/RO water, and cause a significant strain on pre established filter resident bacterial colonies and water stability.
However, if you move over all 12 gallons, that means that you would only be supplementing 50% instead of 75%, and would mean less initial instability within the tank/filter colonies.
Just leaving the sand in the old tank.
Why? Don't do that, move as much as you can over. New live sand is not the same as mature and seeded live sand. Use as much of your seeded live sand (the sand that has been in the 12 gallon) as you can, and mix it with new live sand if necessary to replenish the substrate bed. "New" live sand, is not really new at all, but rather has been sitting in a sealed bag for lord only knows how long. Several hundreds of thousands of micro-organisms and bacterial colonies are depleted the longer the sand sits unmoving with no renewed oxygen or nutrients. Mature live sand that has been thriving in established tanks is far more saturated with beneficial bacterial colonies, benthic organisms, etc. which are all vital little soldiers in a healthy tank. Move as much as you can to the new one.
I would like to keep at least 2 pieces of the liverock and maybe a few crabs in the 12, but not sure if that will work either.
For now I would suggest you move everything over and allow the 24 to level out, as you may see slight fluctuations within the first week. Move the filter media from the 12 to whatever filters you are using for the 24, and let those marinate the new media for at least two weeks before removing them. This will ensure that your new tank does not need to re-cycle, and it will also keep your 12gal filter media seasoned for if/when you want to re-fill the 12 gallon for some rocks/inverts as you mentioned. For now though, I would move everything over entirely, and add a bit more rock since your tank is now bigger.
I would also like to upgrade my 20 gallon fresh, but don't know what kind of lighting is necessary for a coral tank.
Moving a tank from FW to SW is a bit more complex as you will need to completely re-cycle the tank with new filter media, water, etc. But it can be done, and its not "harder" so to speak, just a few more steps. When it comes to lighting, it depends on the corals you would like to keep. Soft corals such as Zoanthids, Mushrooms, Anemones, etc. can do well under High Output fluorescents and T-5's with Actinics, (5,500-8,500K generally)and considering that many of those can thrive in even lower lighting, having enough rock to create overhangs and inlets is always a good idea so they can move where they are most comfortable. On the flip side, however, the majority of other corals will require significantly more, usually Metal Halide.
