Fruits that keep ripening after harvest (climacteric): bananas, avocados, mangoes, papayas, peaches, nectarines, plums, apricots, pears, kiwis, tomatoes, persimmons, guavas, and apples. Cantaloupe softens but does not sweeten.
Fruits that will never get sweeter than the day they were picked: berries of all kinds, cherries, grapes, citrus, pineapple, watermelon, and pomegranate.
Practical rule: most of the first group can be bought firm and ripened on the counter. The buy-firm trick is really for avocados, pears, peaches, kiwis, plums, and mangoes; apples are sold ripe already and only soften from there. The second group has to be picked ripe at the store, so it is worth learning their cues (smell, weight, field spot) before you buy.
Written by the team behind RIPE, a free app that reads a photo of the whole produce pile and rings the ripest ones. Every cue above is one you can check with your own hands, and it all works with no app at all.