Foul Water
First of all, taste the water going into the unit. Does that have the bad taste? If not, then there is something in the lines which needs to be cleaned out. Not sure how you'd do that though.
If the water is just bad, first thing is to put a good filter on the water coming in. This won't purify it, cause you don't want to cut down the flow for showers, etc. However, it will keep out particulates and the one I use (from camping world) claims to filter out bacteria as well.
Next, put a 'purifier' on your kitchen faucet. A good one of these won't flow very fast, but will take care of most bad taste (if it has a charcoal filter), any remaining bacteria, and even viruses. I've got a good one, and it won't do anything for 'softened' water; I can still can taste the 'salts' it added to the water.
If the combination of these 2 filters doesn't give you water you like, don't drink it. Get a 3 to 5 gallon jug or 2 (depending on your strength), and keep them filled with the 'good stuff' for drinking and cooking only.
I get my water from a vending machine outside the grocery store. Only .15 a gallon and tastes great. I've seen similar machines around and in grocery stores here and other places around the country, generally in the .25 to .50 a gallon price range.
There are often services which will deliver water, but they cost more.
And if bottled water is not an option in your area, you may need to go for a real fancy water purification setup, probably using Reverse Osmosis or Distillation. These cost though, both up front and to use (RO seems to discard 4 or more gallons down the drain for each purified gallon you get, and Distillers suck power or gas).