Teaching the basics

Train everyday: It's best to train your dog everyday, even if you only have five or ten minutes, so they won't get bored or forget. If you have a puppy, you should train several times a day.
End on a good note: Try to always end your training sessions on a good note--if your dog has completely forgotten a new command and you're getting frustrated, do something easy that your dog definitely knows (such as the "sit" command) and end the session on that good note.
What motivates your dog? Some dogs are motivated by food; others by praise. Do whatever your dog works best with. Eventually, you'll want to wean your dog off of treats so you won't have to carry them around. But, you may still want to surprise your dog with an occasional treat so she stays motivated.

Sit

A great first command and foundation for more training.

Hold a treat above your dog's head, then slowly move it backwards.
Your dog will hopefully sit in response as they watch the treat going over their head.
If they don't sit, gently push their rump down until he is sitting.
If they stand up, correct with a "no" and place them in a sitting position.
Once they stay in a sitting position, praise and give a treat.

Down

Try with "sit" first - this one's a little trickier.

Start with your dog in a sitting position.
Hold a treat in front of them and slowly lower it to the ground, saying "down."
If they don't go down completely (or at all), gently push their shoulder blades down until they're lying on the ground.
If your dog is very strong, it may be more useful to pick up their front paws and lower them to the ground.
When he stays down, praise give him a treat.

Walking on Leash

If your dog doesn't know how to walk on a leash, the best way to teach him is to simply put a leash on him and walk around. If he is timid or scared, put the leash on inside the house and let your dog just walk around with it on. You could then start to walk him on leash inside your house first, then slowly expose him to cars, etc.
If your small to medium-size dog pulls on leash, it is best to use something besides a normal flat collar, like a gentle leader or Halti. Gentle Leaders and Haltis should only be used temporarily to correct hard-to-control dogs. After you have successfully trained your dog to walk well on a leash, go back to a normal flat collar.
If your large or very strong dog pulls on leash, you might try a correction collar after talking to a trainer or consulting a training book. If used correctly, they should never choke your dog.
Should my dog wear a harness? Your dog should wear a harness if he is very small/has a sensitive neck, has kennel cough that is aggravated by neck strain, or has a tendency to escape from his normal flat collar. Harnesses don't correct pulling. Sled dogs pull in harnesses! There are some harnesses out there that are supposed to help with pulling, but the standard harness is not meant for that purpose.
How do I train my dog to stop pulling? Don't allow him to pull at all. If you don't correct him every single time he pulls, it will be hard for him to realize that you don't want him to pull. You can either correct him by pulling on the collar (pull back gently or pull his head toward you with a halter; jerk the leash firmly but not harshly with a correction collar--make sure it loosens afterwards!) or by stopping (every time he pulls, stop walking so he learns that you will only continue to walk if he is calm).

Stay & Come

Make sure you teach "sit" first for this one.

While holding your hand in the "stop sign" position, say "stay" firmly and clearly. You can repeat the command a few times if you feel that your dog is tempted to get up (but you should eventually only say it once).
Slowly walk backwards, away from your dog. If they get up, correct with a "no" and start over.

When you first start teaching the stay command, you shouldn't go very far. As your dog starts understanding the command, you can increase your distance. If your dog keeps getting up early, try to only make him stay for a few seconds or have someone else hold him in that position.

If your dog doesn't get up early and you are ready to call him, say "[Dog's Name] Come!" with a happy and enthusiastic tone. If he doesn't seem like he wants to come to you, try clapping your hands or squatting down so you're at his level. You can say words of encouragement, but do not repeat the command. Doing so will cause him to ignore you and only come if you call him several times.

Note: It is best to teach the "come" command on leash at first. Place a long leash on your dog so that you can reel him in if he refuses to come. Once he masters the command on leash, you can start calling him when he's off-leash. If he absolutely refuses to come, calmly walk towards him. When you reach him, hold his collar and say the "come" command again. Walk backwards and gently pull on his collar to get him to follow you; then reward him. Never punish your dog if he comes to you (even if he was taking a long time) or when you catch him. It may be frustrating, but punishing him will only make him not want to come to you even more. With a lot of patience and practice on your part, your dog will master the "stay" and "come" commands.