While you can cycle in a pair of trainers you shouldn’t.
When you get the correct pair of cycling shoes your riding comfort and efficiency will improve. Cycling shoes are light and stiff for efficient pedaling, usually have mesh panels and vents to keep your feet cool in the summer, but that does mean that you will need overshoes for when the weather gets colder.
Remember road shoes, in the main, are designed to be used on the bike and not to go shopping in! For this reason, they have very rigid soles, which helps harness pedal power— great for cycling, not so good for walking around.
Well I never…..
In Italy it is the norm to carry a set of covers for your cleats so that you can go into a tiled café for an espresso