Step 1: Let's Gather Supplies!

Let's get this straight. You can't paint your nails without nail polish. It may seem like such a common-known thing, but some people paint their nails with acrylic paint, markers, and wall paint! Eek! Use only the stuff made for your nails. This way it won't damage your nails, leaving them prone to breakage and splits.
You will need:
A bottle of nail polish. I chose a Sally Hansen color in number 390, Red Carpet. It is a gorgeous bright red with differing hues of crimson glitter in it. I bought it at a local Dierbergs, but you can find these polishes at many drugstores.
Nail polish remover: I bought mine from CVS. Try to shoot for an acetone based remover, as it is strongest and removes the polish quickly. Make sure not to let pets or younger humans near it, as ingesting it may be fatal. Keep it away from flames unless you want it to explode.
Cotton balls and Q Tips: Cotton balls will easily wipe off the nail polish if you make a mistake. A Tips will save your booty if you accidentally got a bit of polish on the side of your nail, where a cotton ball lacks control.
A nail file: This is for the longer nailed ladies (and gentlemen!) I recommend a very nice one from Sephora, because the normal sandpaper-like dollar store finds do more damage than good. The $20 is good in the long run also, because you use strong files longer than a cardboard one.
An oil: No, not vegetable oil or Crisco or butter. Coconut, olive, or a special nail/hair blend is what I'm talking about. Especially a must for those who paint their nails regularly. I'll explain why we need it in a few steps.
Optional: Plastic mat. This is so you won't ruin the surface you're working on.

Step 2: Remove Existing Polish

Don't paint over the blue that has chipped off, remove it! It provides much of a neater canvas for your polish, and color blending is often an issue if you keep the old on.
To do this, place a cotton ball to the rim of the acetone based remover bottle, and tilt it upside down. Keep it that way until the ball is damp.
Place on your nail and firmly press down, rotating it in circles.
Dispose of the cotton ball.
Make sure to do this in a well ventilated room, because the fumes are toxic.

Step 3: Wash Hands Thoroughly

Wash hands for thirty seconds under warm water with soap. Buff nails with a towel. You don't want the acetone to remain on your nails or it will weaken the polish.

Step 4: File Nails

Filing nails is actually very simple. Only go in one direction, as not to break the nail. Swift strokes should do the trick.

Step 5: Oils!

We are going to use the oil to replenish the cuticle and nail. With multiple polishes and removers, the nail weakens and becomes brittle. We have to moisturizer and nourish the nails and skin.
Take your oil. If in a bottle, add to a Q Tip and massage the cotton over the nail and cuticle. If spray, spray it over your fingers and massage with your hands.
Use a cotton ball to absorb any excess oil.
The oil I used is actually a hair oil by Organix, called Keratin Therapy.

Step 6: Polish Time

Roll your bottle on your palm and open it.
Swipe the brush on the sides of the bottle rim, to get rid of excess polish.
Paint one thin coat over the nail in three strokes: left, middle, right. It's tempting to glob on the polish, but if you want to go fast and have an even coat, use thin coats.
Repeat three times, or until color is opaque.