My dad always used to take his to the sharpening place, but there aren't any near where I live now. Does anyone do this themselves? What would I use--a file?

Do you not have lawn mower repair places there you live, They usually will do it for you or you could also replace the blades with brand new ones. Plus the place you purchased your lawn mower should be able to help you find some place.

I use a stone on a Dremel tool. Fast and easy. Then check the balance with a round screwdriver mounted in a vise. Lay the center of the blade onto the shaft of the screwdriver and see if there is an excessive imbalance.

Do you not have lawn mower repair places there you live, They usually will do it for you or you could also replace the blades with brand new ones. Plus the place you purchased your lawn mower should be able to help you find some place.

The op stated that there are not any shops near him.
Use an angle grinder or bench grinder, a file will take a lot of time, but gives a nice edge.
Also, VERY IMPORTANT, balance your blade. To balance drive a nail level into a post. Or on the edge of a work bench, after sharpening with the blade clean and free of all debris, place the blade onto the nail with the nail through the center mounting hole of the blade. The blade should hang level, if one side is lower than the other, that indicates that side is heavier and you need to grind more off the lower hanging side. If you use a blade that is not properly balanced, you can, scratch that, you WILL cause damage to your mower. Also, wear safety equipment while sharpening.

I would also add:

Note how the blade is mounted before removing, and remount in same orientation torquing the bolt to the manufacturer's recommended value.
The blade tips tend to round over as they wear. The sharpened edge should be ground back so that the tip is square.
If you notice any cracks or fatigue lines in the metal, the blade should be replaced.
Alternate between edges while grinding to avoid overheating the metal. The metal should not change color. You can use a water pot to frequently cool while grinding. However, were you to heat the blade red hot and then quench it, it will become hardened, and then there is the potential for it to explode if something hard like a rock is hit.
Sharpen frequently. Sharp blades yield healthier grass, better gas mileage, and faster mowing.