In the absence of any special requirements, I'd just choose something popular, because it's likely to have the best support and stick around for awhile. As such, I'd pick from Unity analytics, Facebook analytics or Flurry (aka Yahoo).

I've never used Unity Analytics, but I'd imagine it offers a similar set of features to facebook and flurry and is probably easiest to integrate into a unity project (although the others are probably not hard).

In terms of useful information, I'd expect they all supply a decent amount of demographic information (how many players, what devices they're using, where are they geographically, etc). But a lot of the more actionable information comes from the analytics events you create yourself (which level are people failing too much, which is my most popular shop item, etc), so to an extent, you're in control over whether you get useful analytics information.

There are tons of solutions and providers, with a variety of levels of sophistication.
They can all do simple things like "DAU" and "MAU" and "RPU" and "conversion rate," which are all great first-line metrics.
Most of them can do simple correlations, like "players with event X convert better than players with event Y." Events kinds are typically defined by you.
I've found that, once you reach the point where you want a little more sophistication (identifying and comparing common paths; recommending next actions to users; that kind of thing) then you either have to pay serious money to some provider, or pull it in-house.

Depending on how big your game is, you can fit it all in a single database (Postgres, MySQL, or whatever,) or you'll need to talk to the elephant. Single database is VASTLY preferrable.
Or pay big money to something like Google Big Table or Amazon Redshift.
(There are other providers; I'm not gonna name or shame any, but there once was this one web analytics package that cost my employer > $200k per year, and I spent two weeks of coding and we could cut them off...)

One thing to think about is whether you want to tie your analytics to your acquisition provider (buying web ads, mobile ads, etc), to your monetization provider (selling google ads, facebook ads, etc) or your platform provider (if you live on something like Parse, R.I.P.)
Choosing an analytics package that integrates well with your engine often seems like a good idea from ease of implementation, but it turns out that even the hard ones are easy to implement -- you need to identify your game to some SDK, you need to identify the user using some username/id, and you need to insert "X happened" calls in your code.
The real differentiation comes on the back end; what analysis do you get; how do you get access to your data if you want to take it elsewhere; how long do they keep your data; etc.

Unity Analytics? Mostly Harmless, somewhat useful, probably not enough once you have a real, working business.

As with everything else, though, details matter, and making sure you really do understand your requirements up front is important.
If you haven't done web and acquisition and monetization analytics before, you're unlikely to know what you ACTUALLY need, so be prepared to change horses once you learn more.