Those bubbles are nitrogen, which apparently can sometimes cause a problem to fish by gathering under the skin or in their organs. I notice a LOT more of it in water from the hot tap than just the cold.
Just to be on the safe side, I like to 'age' my water for at least a day in VERY well rinsed (no detergent) milk containers. Most of the nitrogen's gone the next day.
I remove my betta from his tank, to a large ziploc bag with some of his old, warm water (then I sit the bag in a big yogurt container for stability) before adding the cold, aged water to the tank.
Then I turn the heater back on and wait the 5-10 mins or so for it to heat the new water to the right temp. Then I 'float' Sid in the bag for at least a half hour before putting him back in the tank.
This causes my fish little to no stress, as far as I can see, and doesn't take very long at all. I add the conditioner to each container of water according to the drops-per-liter directions before pouring it into the tank.