Keeping it simple!
There are many water filters out there, from pitchers to RO membranes to charcoal sticks. I have played with a whole bunch, and my favorite kind of filter is this one:
The 1 Micron Carbon Block Filter
It hooks up directly to the end of a faucet. You remove the aerator and screw it on. Carbon block filters also can come as fridge filters. The activated carbon removes chlorine taste and nasty chemicals from the water, leaving the minerals intact. So, if the mineral balance of your tap water is decent, this might be the best way to powerfully clean it up without taking anything beneficial away.
UPDATE: I now prefer the waterdrop brand filter pitcher, it’s slightly better than these carbon block filters and certainly a bit easier.