The "every 3,000 miles" oil change is a hold over from the days when all motor oil was simple mineral oil. If you still run straight mineral oil (non-synthetic) then yes, change it every 3,000 miles or six months whichever comes first.
If you run synthetic engine oil, then it's different. 5,000 miles is easy, probably 99% like new oil still. 8,000 to 15,000 miles is the kind of range you can have on synth oil. It's hard to say how often you should change synthetic if you don't have a true oil life monitor built into your car (like the ones from Cadillac and Mercedes, they're amazingly accurate). Very safe rule of thumb is 6,000 miles, your oil WILL be fine at that interval.
Also, oil change places want you to change your frequently, don't they? So the only guide they use is 3,000 miles regardless of whether you're synthetic or mineral.
Brief rant--USE SYNTHETIC! It's so much better for your engine. Unless you're on a 2 year lease and don't care, your engine WILL last longer and run better on synthetic oil. Use synthetic, change oil and filter every 6,000 miles (except for the for the first and second changes, ideally they would be around 500 and 3,000 miles) and your car will purr dang near forever.
By the way, changing more often can actually be bad for your car to say nothing of the environment. The worst wear your engine gets is the first 15 seconds after an oil change.
As to other maintenance, read the manual. Follow the normal or severe duty schedule based on the description of these in the manual. Since I drive hard and our cars see a lot of city miles, I use the severe schedule.