They keep on the tree for a couple months.

The best way to tell if citrus is ripe is to sample the fruit. There is usually enough fruit on a tree so it isn't a big loss if a few are sacrificed by sampling. If you keep records, after the first year you should know when it is
time to harvest.

You can also get it a ball park estimate of the right time of the year but you need to know what varieties you have. Some AZ Sweet varieties ripen in November but Valencias won't be ready until February-March. See http://cals.arizona.edu/pubs/garden/az1001.pdf for some examples.

Once ripe, the best place to store it is by leaving it on the tree until you are ready to use it or until it becomes overripe and/or drops from the tree.

My question is whether oranges that have sat on the soil for a day or two are OK to consume. Ones that have been on the ground for more than a couple days seem to get moldy quickly, and so to avoid turning the orange trees into fungus labs, I'm currently picking up all fruit on the ground and throwing it out. But if a fruit on the ground still looks good, is it OK to eat?