The only sure way to test how well a water filter works is to test the water that comes from it. Hardware stores and home centers have inexpensive household test kits available, or you can contact your local water utility for the location of a state or national testing lab.
Which filter is best for you depends on the level of filtration you want. For taste, odor, and particles, a basic activated-charcoal filter is enough. For certain water-borne parasites, heavy metals such as lead, and other contaminants, you may need a reverse-osmosis system.
Before you can decide which type of filter is best, you need to know what impurities are in the water. Then you can get a filter that's specific to those impurities.
Yes and no. Reverse-osmosis filters remove just about everything, both good and bad. Charcoal filters don't remove fluoride but may not remove other undesirable contaminants. It's important to find the correct filter for the contaminants you want to filter out.
By law, to be considered safe (potable), municipal/city water must meet strict guidelines. You can find out more about the water you drink by contacting the municipality that provides it. Even though your city water is safe, you may want to filter it to remove tastes and odors that you don't like.