I’d stick with the stock tyres for now, but like with a mountain bike, road bike tyres make a significant difference to things like ride quality, rolling resistance, grip etc. I’m constantly amazed at how big that difference can be.
After a recommendation from Strada I recently fitted a pair of Vittoria Open Paves and they really lovely. The bike rides better, feeling like there is less resistance and greater comfort, sort of like the difference between a set of clinchers versus a good set of tubulars.
The Open Paves are the clincher version of the tubular tyre that is designed for use in things like the Paris-Roubaix, i.e. cobbles, mud etc, so they have some puncture proofing also built in. They aren’t cheap mind, but they are quality.