when the author says that the new boy wore shoes on Friday

We never find out the boy's name here. Twain calls him "the new boy". Nice clothes of any kind were an anomaly for boy's like Tom. The only day Tom wore his best clothes, and even they were not very good, was on church Sunday. It was Friday and, "He had shoes on — and it was only Friday. He even wore a necktie, a bright bit of ribbon." The two boys end up fighting for no particular reason.