At first, a 700x25 equipped bike (meaning 25mm wide tires - your new bike) will feel a bit less stable than a 700x35 equipped bike (35mm wide tires, typically what you find on a hybrid). As NY says, as long as the bike fits you, I'd say just do a bit more riding. You'll get used to it. The 700x25 setup is actually pretty durable. Most of us have put umpteen thousands of miles on 700x25 road bikes on pavement of all kinds and even gravel and dirt at times. They're not as dainty as they look.

The most dangerous cracks in the road with skinny tires are the ones that run parallel to the road, the ones that point in the same direction as your wheels. These can pinch your wheels if your wheels fall into them and cause you to lose steering. These are dangerous. The much more common horizontal to your wheels cracks and shallow/small potholes a 700x25 can handle pretty well. Not the most comfortable, but the closer you hit them at 90 degrees, the better.

As to going to a bigger tire, looks like the Giant is a pretty standard road bike. Most of the time, you might be able to squeeze a 700x28 on such a bike, but not always. Take your bike to the dealer and ask them, but as I said, I'd jut do a bit more riding on the 700x25s. Put some more miles on them and they'll come to feel very normal.