Your fig tree can have two harvesting times during the growing season, in the spring and fall. Usually the latter is the heavier crop. You should only pick figs that are fully ripe. Figs will not ripen once they are removed from the tree.
When is it Ripe?
Figs are ripe when soft to the touch. You should also be able to look at the fruit to see if it's ripe. The fruit will start to bend from the branch, almost drooping. Another way you can tell is that the fruit will gently come off the tree without much help. If it stays on the tree and requires tugging or cutting, it's not ripe. If you collect the fruit as it falls to the ground it will also be ready to eat or be stored.
Next Year's Harvest
Part of each year's harvest is to prune away the old wood from the season. You should do this immediately after picking your main crop. If you don't do it immediately, it's better to skip the pruning until next year versus a winter pruning. You will not give the tree enough time to heal and get ready for the next crop.
Fruit from a fig tree basically comes to you when it's ready. Make sure to set aside the time to harvest your fruit so it doesn't go to waste on the ground. If this happens you can mulch the fruit or see if there is any use for overripe fruit in your area.