Should I worry about this? (does it reduce the effectiveness of the shingle). If yes, then question 2 come into play.
There's certainly more important things to worry about in life, but, yes, Moss will, over time, shorten the lifespan of asphalt shingles.
Can it be corrected in the area that are truly unsightly with a chemical or something? I have kids so it would need to be safe. If yes, question 3 comes into play.
Yes. Actually, you want to correct it over the entirety of the roof. You can purchase spray-on as well as granule based moss killer. That is what you want.
If some cleaning can be done to fix, is this a DIY or does a service have to be contacted?
You can DIY or you can hire a service. If you live in a moss-friendly region (such as the Pacific North West of the USA) then you'll find all sorts of people that do pretty much just this. Either way, be sure to hire people that use very gentle cleaning methods. You don't want power washing or anything of that sort.
Once corrected, is there any preventative measures that can be done to avoid this?
The only real remedy is to not shingle a roof with asphalt shingles in moss-prone regions. That's why you see a lot of standing seam metal roofing in wet areas. Barring that, you just need to make the moss prevention an annual chore.
Some claim that zinc strips will work (as zinc will kill the moss) though, at least around here, you can find plenty of roofs with zinc strips that also still have plenty of moss on them, so I tend to think that's a bit of a scam solution.