Tap Water ok to drink in Italy? In Barcelona?

It is fine. These are not third world countries.

Both cities are fine. Public fountains in Rome will have signs “potabile/non-potabile” . In Barcelona anywhere is safe. Never had any issues with water anywhere except Egypt.

Rome has hundreds of "nasoni" (meaning big nose) fountains which run continuously and have delicious water to refill your bottle or drink on the spot by pressing your finger over the tap. Don't hesitate to use them.

Not third world country but almost all of my friends here in the Modena area do not use the tap water. I do but everyone else buys bottles water. Rome has fountains available that you can fill up your bottle.

Water is safe everywhere, unless a public fountain has a sign explicitly saying "Non Potabile".
Rome is famous for having good water without the slight chlorine taste that one might notice in other cities (including my own in California).
In any case bottled water in Italy is super-cheap (as low as 25 cents for 1,5 liter bottle at a grocery store), so you can also buy it without breaking the bank.

https://vividmaps.com/tap-water-safe-to-drink/amp/

My wife and I have gotten into the habit of buying a 5L or 6L bottle of water from the grocery store. The bottle generally costs about 2€ and we refill our smaller bottles from it. Since we normally have a rental car the bottle is kept there. If we’re staying in a city with the same lodging for a few days we’ll mostly use tap water. Also, rather than taking water bottles with us, when we buy a bottled soda or drink, we’ll just refill that bottle with water.

I will reiterate the point about ensuring water from fountains is potable.

Thank you everyone! I love this group!

Safety is guaranteed in both cities. Something else is taste. In Barcelona, tap water mainly comes from the Llobregat river which passes thru calcareous mountains and "catches" a bit of a flavour (google "hard water") which is fine for 99% of people but some might be put off.