In most cases, adhesive tape, alcohol wipes and gauze pads will do for the majority of scrapes from riding a bike. On the rare occasions where an injury may be quite a bit more serious, it may well be that first aid knowledge is called for. In most places in America, in the event of serious injury, emergency personell with much training with a truck full of special equipment will be on the scene very quickly. There are times and places where that would not be the case.
Some years ago during serious back packing days, I took a two day first aid course through the Appalachian Mountain Club. This course is highly regarded in this area and in fact there were two MDs taking this course to pick up tips on how to handle serious injuries when help is far and many hours away. I would suggest that the contents of a first aid kit is only half the solution to injury. The other and equally important half is some knowledge of what to do with all the stuff in the kit. I would recommend people look for First aid courses, such as from the Red Cross which are conducted in many parts of this country.