I know you can put pears in a brown paper bag and leave at room temperature. It may take several days.
Tomatoes and such I just leave on a window sill. Never put a tomato in the refrigerator unless it is ripe enough and you aren't going to eat it for several days. But a sunny window sill will continue the ripening process.
Plums, nectarines, peaches can be done the same way as tomatoes.
A cantelope can be cut in half, the cut half placed on a plate so that NO AIR enters around the bottom (so the melon and the plate have to be perfectly flat) - in a matter of just a couple days the melon will be perfectly ripe and sweet.
Hope this helps. An avocado can be ripened in a paper bag also.
The above advice is correct, but keep firmly in mind that, excepting pears, none will develop the true flavor and texture of tree ripened fruit. Pears, oddly, improve in flavor and texture when ripened as described, or merely on a window sill.
It is claimed that adding an apple to a paper bag of unripened fruit will help - the apple emits ethane gas, which is used commercialy to ripen fruits, etc.
I find that giving close attention to the "feel" and aroma of fruis in the store can result in bringing home better quality - often even fullly riipe. Thisworks especially well if you know when the store receives the fruit, and be among the first customers to purchase it.