Yes, if you have the OTO kit you can start by testing you tap water with that, so we know how much (if any) chlorine is in the water and its pH. Sometimes you can find the WalMart kit that also tests Total Alkalinity, Calcium Hardness and CYA (stabilizer) . If so, test the TA and CH, but not the CYA (they don't put that in city water).

Personally, if I knew NOTHING about the tap water, I'd add one gallon of regular, unscented bleach for every 10,000 gallons, as it's filling, just to prevent anything growing while you are starting up. About the only thing bad that can happen is if, by some weird chance, your city water has a lot of metal in it, there could be a problem. But metallic water is usually well water, not city water.

But a full readout could be very helpful in tailoring treatment for your pool.

You mentioned something about not liking a lot of chemicals in your pool. Unfortunately you won't able to escape very high levels of the most dangerous one by far, Dihydrogen Oxide, sometimes referred to as Hydrogen Hydroxide.

Carl