You'll need to pitch the correct amount of active yeast for a quick start to fermentation. Yeast will have a lag phase as it wakes up, then an aerobic growth/lag phase as it reproduces, then an anaerobic fermentation phase followed by a cleanup phase. Assuming you are pitching just a normal home brew pitch of yeast, such as one vial or smack pack, but different wort gravities, the one with the most fermentability should theoretically take a little longer to get started, because it should need a longer growth phase to get the yeast cell population high enough. Yeast can sense the amount of fermentability in wort, and will continue reproducing in the growth phase before it has enough brothers and sisters, then begins fermentation.
To the second part of your question, if you have a quick and vigorous start to fermentation in the first 24 hours you are doing something right. Final gravity and apparent attenuation are strain dependent, but you should be able to maximize it to its potential by keeping the yeast happy. That means pitching an active and healthy yeast OR pitching healthy yeast into oxygenated wort where it will reproduce, keep everything at the optimal temperature, controlling your temperatures, and having yeast nutrients in the wort. You could go for the high end of the temperature range for your yeast strain for maximum attenuation but that may introduce off flavors.