Though I could write dozens of pages on the subject of water, I will try not to deluge you all in a flood of philosophical musing. That said, get out your umbrellas!
Back in 2000 (or was it 2001? 1999?), I remember Benjamin of Rishi Tea fumbling around under his demonstration booth at CoffeeFest muttering, rather vehemently, "They told me this was good water!" He had several large water-cooler-sized jugs of water under a table; I think he was trying to strangle one when I walked up to his booth. "This water sucks!"
I had to almost fight to get him to make a cup of tea; he was too ashamed of his tea prepared from such disappointing water. After many disclaimers and qualifications (which, naturally, made me smitten), we brewed some Oriental Beauty gongfu style. In a word: Amazing. Even with subpar water. Simply sipping in his fuzzy-faced company was swoon enough for me. The care, the grace, the focus he brought to the preparation heightened each layer of flavor.
When I brewed some at home later, it wasn't nearly the same near-religious experience. Oh, I had good, fresh, clean, soft, dechlorinated, purified pricey water. I just didn't have the grace, the honor, the enlightened company. Nobody to say, "Do you taste that? I feel a circle of vanilla rising!"
*sigh* Well, yes, water does make a difference too. Bottled soft spring water with the proper PH and the right mix of trace minerals is ideal except for the bottling part and the transportation part and the shelf-life part and the cost part, not to mention the part of lugging it from store shelf to kitchen counter. (Hey, I drink A LOT of tea!) Did I mention the cost part?
Brita filters and Pur filters and the like can help immensely. The only problem with them is that they truly don't last very long--I'd advise their real life in terms of tea preparation to be less than half of the proclaimed lifespan for general use. If you use them, it's worth actually counting how many gallons you use them for and replace them on a strict schedule.
When I used them, I actually double-filtered my water. First, I put water through a faucet-mount system which I used for all my drinking and cooking water. Then I put water meant for tea through a second refrigerator pitcher filter. This allowed me to change the general-use filter less frequently and the tea filter more frequently. The quality difference versus one filter for everything was significant.
Even with coupons and sales and discount clubs, my purchase of water filters became quite expensive and just a bit of a hassle. So I consulted with the proprietor of a local teahouse I respected and he referred me to a commercial plumber who specialized in water purification for coffee and tea establishments in my area.
I eventually had a semi-professional under-the-sink system installed which meets all my water needs for a little over $500 for the first year, plus a smaller fraction of that on-going from there. It's still cheaper than bottled water, about the same cost as countertop or faucet-mount retail units. But it made me make a commitment and never look back.
I used to say, "Maybe I can run this one more teapot . . . one more day . . . one more week." I would slack on changing filters or buying the best bottled waters. I had to justify the indulgence to myself time and time again. Even though I had made up my mind that the indulgence was worth it, I still had a hard time following through.
Also, I had used much better water for tea than regular drinking or cooking or even coffee. When I switched to a semi-professional filtration system, I noticed that I drank more water in general, not just more tea. I enjoyed water more. Life had become just a little bit better in a small yet important way.
Anyway . . . I think it's good to filter water at home instead of buying it in the store. I think it's worth consulting with someone who has experience with your particular water source to see what's feasible and most important and within budget. (Options range from reverse osmosis to carbon-block filters to granular carbon filters to activated alumina to anion and cation exchange, etc.) I think it's nice to occasionally buy artesan and spring waters and taste-test compare them in making tea as an educational experiment as well as entertaining exercise. And yes, I think tea can be highly enjoyable even if your water sucks.
Remember, tea became so very popular because humans could not drink water for many centuries without boiling it first. The bacteria and scum and filth in water were widespread rampantly worse than we could dream today. Then again, tea lovers of centuries past would never have dreamt of the perils of chemical water "treatments" and the fall-out industrial pollutants we have on offer today.
Distillation is a very effective filtration method; it's a great killer of offensive taste and a great killer of taste, period.
As for measuring Total Dissolved Solids, a number of manufacturers make a TDS Meter. Unfortunately, I don't think this will tell you what exactly those solids are, but it will give you the sacred ballpark count to let you know if you're in the zone at 10 to 30 parts per million.
Though Dupont swears it's completely safe, the presence of Teflon in drinking water disturbs me. It's been found in both tap and bottled water supplies. It's also turning up in greater and greater percentages in the hospital delivery room, detected in higher and higher concentrations in umbilical cords. I guess we're entering the era of non-stick babies.