You need tire levers, a multitool and a pump for any ride that is longer than a couple of miles if you don't want to risk being stranded. Historically for me tires have tended to flat at the least opportune moments (in the middle of a century ride, for example).

A rack is not strictly essential, but it sure felt that way for me because I hated wearing a backpack. You don't need panniers or a trunk bag if you don't want to spend too much. You can always bungee a regular backpack directly to your rack, or attach a milk crate with zip ties. Not as 'sexy', but very functional.