I've been researching this question on line, and as best as I can tell, Brita does NOT filter out calcium. Or at least, their website is notably silent on the topic, and other articles have said things to the effect that Brita filters are essentially good only for cosmetic reasons, in that it filters out things that make water taste bad, like chlorine, but little else.
It does not filter out bacteria, which I'd always assumed it did. Quite conversely, I read an article that says if you don't change the filter regularly it can actually grow bacteria and be worse for you than tap water.
Another thing I'd always assumed: that it would filter out lead. The standard brita does NOT filter out lead. You have to get the higher quality one, called longlast.
What Does Brita Filter Out? Chlorine & More | Brita®
It looks like a reverse osmosis system may reduce calcium in the water, and is effective at removing bacteria.
Also, if you just want to remove calcium, then a water softener will do that, but it uses sodium to do it (exchanges the sodium for calcium and magnesium) which then causes saltwater waste which causes a lot of other problems. For that reason, it has been banned in several areas in the US.
My personal conclusion for best solution both for safer water and reduced calcium: reverse osmosis system.
Disclaimer: I am not a water expert, so if someone else is more knowledgeable about this, please do weigh in.