I'm an amateur genealogist, and I think there are several ways to look at this, or rather, to tackle it. One is to join a site like Ancestry and put in any info you already know (names and dates, mainly), build your tree as much as you can, and then look for member trees that are a close match (that's not hard to do, as the site will do that for you pretty much). When you find members with matches, you can message them and ask for any additional info they have. This is easiest when you at least know some ancestors that have passed away (as living individuals usually are harder to pinpoint, since their info is usually hidden from public view). You would be surprised how much you can find on there even if you start with just a few names (pull any info from public social media as you can to fill in gaps; I find cousins by sleuthing through various people's public friends lists on Facebook, for example). There are members on Ancestry that are really helpful, and some you'll message and never hear from...
If you go the DNA route and only get the results they send directly to you, it's mainly just going to give you an idea of what percent of the ancestors are from where. It's not going to reveal stories, or histories really, just geographical data. However, if you do the DNA test, like through Ancestry, and merge it with your family tree, you'll then be shown public matches of individuals who have also done the DNA testing and linked through the trees. If so, you can contact them, and they can contact you, and hence get a lot more info and history. There's obvious pros and cons. Then, there's the other, more medical tests (like 23&me), but I don't know, nor follow much about them.
If it were me and my kid (I don't have kids, so this is all hypothetical lol), I'd take the DNA test myself (if comfortable with it) and wait til the kid is at least 18 if not 21 to let them choose to do their DNA. There are privacy ramifications... though I think we are kind of fooling ourselves as to privacy if we have had anything at all medically done that removes any type of tissue from our bodies, that is usually always sent off for some type of testing.
There used to be away to have the DNA test done and then request the sample be destroyed so at least nothing further could be done with it, but I haven't kept up on that. Any test you choose should offer you the data, which you can then do different searches with through something like DNAGedcom, which further analyzes your data by running it through a huge database, again, not something I'm familiar with, but people say it makes the info a lot more specific as to things like geographical locations of ancestors.