Bottled water is a pretty interesting thing, as there are a few different types out there these days, all of which is regulated by the FDA. Tap water is regulated by the EPA.
So long story short, depending on what type of water it is, there are different types of minimum requirements to meet. Bottled water is typically, “cleaner” than tap water, but only because it goes through some additional treatments.
For instance, spring water is the same as tap water because of the way it is transported and stored at the bottling facility. However, before it is bottled, it goes through carbon filtration to remove the chlorine. Then it goes through a micron filtration process to take out the floating particles, and then finally an ozonation process to kill bacteria and other microorganisms.
Mineral water is similar to spring water, but bottled water labeled "mineral water" must contain no less than 250 parts per million of dissolved solids, and cannot have added minerals. Each type has different levels of minerals, so check the nutrition label to verify the mineral content.
Filtered water is also similar to spring water, except the source is municipal tap water. It goes through the same carbon filtration and ozonation process, but it may or may not go through a micron filtration process.
Purified water is water where the impurities must be removed or reduced to extremely low levels. Water which meets this definition is of higher purity than spring water, tap water or filtered water. Minerals are removed by one of several processes, including distillation, reverse-osmosis and deionization. Purified water is also known as "demineralized water," "deionized water," "distilled water," and "reverse osmosis water."
So there is no reason that you can’t get the tap water at home to be as good as or better than certain bottled waters.
Also, the benefits to drinking mineral water is somewhat minimal, as you can get the same minerals – calcium, magnesium, and sodium – from eating different types of foods.
There are a bunch of types of water filtration out there - Distillation, deionization, carbon adsorption, ultrafiltration, reverse osmosis, UV, and infrared.
So do a little bit of research and figure out what type of things you want to have removed from your drinking water and go from there.
All of that being said, I currently use this in my home:
https://www.amazon.com/PUR-18-Cup-Water-Filter-Dispenser/dp/B0006MQCA4