I use a DIY version of a Berkey filter system and it tastes like absolutely nothing. I'm super sensitive to tastes and smells and I have to flush filters often 4 times or more than the recommended amount in the instructions. If you have flushed more than a few gallons through and still have poor taste then there may be a problem. I've seen some tests suggesting that there are different filters in the same price range that are better at filtering particulates from water, but I'm not exactly certain which particulates were getting through or how it affects taste. In my case I am currently filtering
city water and it works well for me.
If/when I to move to rainwater or well water I would want to test the filtered and unfiltered water before using it and with some sort of regularity. I was just listening to a podcast where they used their well water for irrigation, and filtering only your drinking water while eating from a garden watered with something unhealthy could be harmful.
By filtering everything out of your drinking water, it removes minerals which can cause problems for some people if those minerals aren't replaced with another source. Clean natural springs only exist in select areas in my region, so without access to that I prefer to filter everything and find other mineral sources. Since you are what you eat, or drink as the case may be, it might be wise to not just test your water, but also get your blood tested and bring up the water situation with a doctor.