You certainly can grow plants from grocery store fruits. Stonefruits, such as cherries, plums, apricots, and peaches will require cold-stratification and should be Winter Sown (see the forum and it's FAQs for how-to's), apples and oranges do not need cold-stratification and can be sown into cups and placed in a sunny windowsill. An avocado is a very good first-time project for this as the seeds are very large and easy for young hands to work with.

Trudi

Here is a link that might be useful: Grow you own avocado tree