Apart from a functional bicycle, the rest is open to debate.
Anything longer than a post-adjustment test ride, I carry stuff to fix a flat. Tube, tire levers, pump. Self-adhesive patches.
Anything out of easy walking distance I also carry a folding multi-tool for bicycles, incl a chain breaker and a quicklink.
Then I keep adding, depending on length/remoteness of ride. A snack, fluid, small first aid kit, maybe a windbreaker jacket.
I really like my bike shoes. And the padded cycling shorts. Jerseys work fine too. Helmet seems like a good idea, might be an easy way to avoid a split eyebrow, if nothing else.