I'm pretty sure it has R/O right on the label. I bought about 20 gallons awhile ago. You can also use distilled water, it is different but still good.
If you use the same LFS, and they know your face. It is quite easy to take them a sample of your tap water and ask them to test for phosphates and nitrates. Depending on your city, you probably should ask them to test for copper too. If you are going to have invertabrates (shrimp, corals etc) you cannot have any copper in your tank.
Just be patient with your tank cycling. The only thing you should be testing yourself are Ammonia, nitrites, pH and salinity. Ammonia should hit 0 pretty soon, nitrites will take awhile. You can probably relax and only test once per week. Your nitrates will stay low for awhile. What is happening is you don't have the bacteria necessary to make nitrates yet. That is why you have ammonia and nitrites. I would not even bother testing for nitrates until your nitrites are 0. Your fish do not produce nitrates... your fish produce ammonia, which gets turned into nitrites, then another bacteria turns nitrites into nitrates. You need to get all those bacteria growing until you see nitrates. After your tank has cycled, then you need to figure out how to deal with nitrates. One method is water changes (5-10% monthly), another is macroalgae.
After that, it really depends on what you plan to do. I am back to using tap water because it is relatively safe, with only trace amounts of phospates. However, I have a 20gal refugium that has a ton of macroalgae growing in it. The algae removes the phospates from the water, along with nitrates. The more diversity you have in your tank, the more you can get away with.