How long will my engine oil last in my car?

I respectfully have to disagree. The 3,000 mile oil change was a good idea 30 or more years ago but in modern cars with modern oil and filters it is a total waste of money and oil. There are still a few car models that recommend 3,000 miles but they are rare. Only oil change shops who make money on oil changes recommend the 3,000 mile oil change as a marketing plow to get your money. Almost every auto manufacture recommends 5,000, 7,000, or even 10,000 miles. Of course it also depends on how you drive and where. If you drive in very dusty conditions then 3,000 miles might be appropriate. But for the average driver that is just overkill and a waste of your money and oil. Change your oil according to what the manufacture of your car recommends and do not listen to people who make money off oil changes or people living in the past.
Optimally, engine oil should be changed every 3,000 miles. Some car manufacturers have modified that to 5,000 miles.

Engine oil breaks down after a period of time as well, so it's best to change that oil at least every six months to a year, even if the vehicle hasn't gone 5,000 miles.