In practice - if you want to wait for less than millions of years, fast growing organisms are likely the only practical solution. It is unlikely that an existing organism is optimal for any of the steps I suggest below, so almost certainly the organisms would need to be extensively engineered. Alternatives, such as big mechanical factories pumping out O2 doesn't seem at all realistic.
First though, I think it is worth commenting that any conceivable terraforming process is likely to take many millennia to produce a livable stable atmosphere/climate. But with that niggle over, I think that an efficient solution may be a multi-step process. The exact process would obviously depend on the starting point environment, but as an example, take something a bit like a possible primordial earth: N2, CO2, a bit of NH3 and CH4 and H2O, but no free O2. Assume that most of the H2O is vapour as all that CO2 produces a moderate greenhouse effect.
Step1: Seed with single cell organisms that mop up the ammonia and methane, and breath CO2. The atmosphere may be fairly opaque so photosynthesis is not necessarily an op-tion - organisms might 'eat' sulphur or use some other novel metabolism. Design them so that when they die a significant part of the CO2 they have respired remains 'fixed'. For example, they produce a carbonate shell, or just coat the ground with a growing peat-like carbon sink. Because there is no competition in the environment, these organisms will multiply to form a significant biomass only limited by available space. 258
After a long time...CO2 is significantly down and the planet cols sufficiently for precipitation to occur.
Step 2: Probably more CO2 breathers but now aquatic ones. Potentially more efficient and easier to sequester carbon in deep oceans. Much more time passes....
Step 3: Add robust O2 producers. The atmosphere shifts from reducing to oxidizing, so there will be enormous changes in rock weathering, mineral deposition etc. Hopefully the air is clear enough now for photosynthesis of some form to be the main mechanism for organism growth. More time passes...
Step 4: Now sufficient O2, but may need to add/subtract various organisms to ensure that you don't go too far either way in green-house effect. There may be periods of high temperature and/or a snow-ball earth effect that you have to manage by careful injection of organisms that raise or lower green-house gasses or alter the albedo of the planet. This won't be just a simple formulaic solution - active management will be required.
Step 5: You now have a planet with a limited number of single cell organisms that is not too hot or cold, surface erosion has more or less stabilised, and you have an at least approximately breathable atmosphere. In a number of planned steps, start introducing macro-plants and the first animals (initially microscopic). This will need very active management as any imbalance could produce a plague of frogs effect or an unwanted extinction.
Step 6: Now you can possibly introduce you colonizing community. The environment is still probably miserable to live in - too hot/too dry/too cold/too wet/hurricanes on a weekly basis/too much UV/everyobe eats tofu 'cos that's all we can grow/but the settlers' great great great.......grand-children might have a comfortable life in a few more millennia - if you can maintain and slowly improve the necessarily knife-edge balance in the environment 279