Avocados don't ripen on the tree ... they ripen after picking, which is why timing matters so much once you get them home.
How to tell if it's ripe
Gently press near the stem end, not the middle. It should give slightly under firm pressure. If the stem cap is missing and you see brown underneath, it's likely overripe.
To ripen faster: paper bag method
Place the avocado in a paper bag, optionally with a banana or apple, and leave it at room temperature. The ethylene gas these fruits release speeds ripening, usually within 1-3 days.
To slow ripening down
Move a ripe (but not yet soft) avocado to the refrigerator. Cold slows the ripening process significantly, buying you several extra days.
What doesn't actually work
Microwaving an unripe avocado softens it through heat, not ripening ... it will taste starchy and off, not like a properly ripened fruit. There's no real shortcut around time and ethylene.
Storing a cut avocado
Leave the pit in, brush the cut surface with lemon or lime juice, press plastic wrap directly onto the flesh, and refrigerate. It will still brown somewhat, but far more slowly.