I am considering doing a second fermentation to clean up the taste and look of this beer.
Do you mean a second
ary fermentation? Secondary fermentations aren't actually a "second" fermentation. They are in most cases just an intermediate conditioning step. There's an active debate - that I won't rehash here - about whether they're even necessary or appropriate where you aren't adding additional components (e.g. dry hops, fruit, and candi-sugar). I think that I can paraphrase the general consensus by saying that, if your yeast is healthy to begin with, you can achieve (basically) the same results by leaving your beer in primary for an extra week or two, and avoid the possibility of infection by moving to secondary.
If you really mean you want to start a
second fermentation, then I'm just plain confused. You want to add new yeast? That won't clean up the taste and look of the beer; it will create a
different beer. Which may be better than the first (I blend yeast all the time) but won't automatically be clearer.
What should I look for when initiating the second fermentation? Decrease in airlock activity? A desired hydrometer reading?
You don't have to "initiate" secondary fermentation, because it's not actually fermentation. If you
want to secondary your beer, you can transfer the fermented beer from your primary fermenter to a (sanitized) secondary fermenter. The purpose is to take the beer off the yeast cake which has formed at the bottom of the primary fermenter.
How much actual fermentation occurs in the second stage?
Unless you add additional sugars, none.
Should I look for additional activity in the airlock or wait a few days and check the hydrometer readings?
Unless you add additional sugars during secondary fermentation, the yeast won't have anything new to consume. Because of this, you shouldn't see any change in your hydrometer readings. You
may seek airlock activity, but that's caused by the beer off-gassing suspended CO2, or by changes in the ambient temperature and air-pressure.
And how long should I wait until I bottle condition?
As I noted earlier: some people, myself included, don't go through a secondary step at all. If you believe a secondary stage is appropriate for your beer, then the time you spend there is dependent on your purpose. If you just want to clarify the beer, then a week is probably more than sufficient. If you want to dry-hop your beer, then I usually wait two weeks. I'll often leave beer on fruit for three weeks (because the fruit
does add new sugars, and thus re-starts active fermentation.)
It sounds like you just want to clarify your beer. Sometimes people who use a secondary fermentation for clarification refer to their secondary fermenter as a "bright tank." I like that term, because it's less confusing than "secondary fermenter" - a vessel in which no fermentation takes place! If that's your plan, I'd leave it in for a week, and then bottle or keg. If, after a week, you're not happy with the clarity of your final product, you can wait a little longer.
You also mentioned that you wanted to "clean up the taste" of your beer. That's often referred to as "conditioning." The yeast in suspension in your beer will continue to condition the beer until they die or become dormant. They can do that just as well in bottles as in a secondary fermenter. So if you're happy with your beer's clarity, it's time to bottle. The yeast will take care of the rest.