Make regular body examination a regular habit with you and your companion. Check the eyes, ears, nose, mouth (teeth), back, back legs, tail, underbelly, front legs, and chest, take your dog for your regular vet check ups.
Most importantly "make this a positive experience for you dog".
 
Look at your dog: Perform a Daily Exam 
 while sleeping; walking around; eating; and/or performing other activities.
 Get to know your dog, by doing a daily exam.

NOSE TO TAIL TO TOES.
 Look for any physical changes:  Start at the head and work your way down the whole body by: 
 the way  your dog walks, runs; how the dog sits; or lies down

LOOKING; LISTEN, FEELING. Are there any:
 
Lumps; Bumps; Irregularities.
Does the dog move away when you touch a certain area?
Does he yelp in pain?
Will he accept your touch everywhere on his/her body?
Check the eyes, are they bright and shinny and full of life?
Are the dog's teeth clean, and none are broken?
Smell your dog's breath.
Check the Head & Ears (ensure the ears are clean, and do not smell):
Run your hands down and around the body even down and around the tail.
Check the condition of the dog's coat.
Check your dogs feet: are the nails to long? (ensuring to check between the pads of the feet.)
When the dog urinates and defecates, watch the dog to see if there is any straining or discoloration

 If you are training you can stop off at the vets with your dog walk around praise him for appropriate actions and leave. The dog only goes to the vet once a year and most times he is not comfortable there as in his mind something always happens to him.  
 
Look into breed specific problems, and find out what to expect.

Learn how to give your dog liquid and solid medications.
 
 When you are training or out with your dog ensure you are aware of you environment. Broken glass, uneven ground or potholes, any wild animals in the area etc 
   
 Learn what to do in case of an emergency: Stopped breathing, heart stopped, poisoning, wounds and bleeding. Many times in an emergency seconds count.

  Create an exercise program for you and your companion age and ability for both owner and dog will dictate what you and your companion can do.
 
 Find a veterinarian that you are comfortable with and learn from him, if you do not know ask your vet or do some research on your own. Remember not everything we read is the truth. Find out the source for the material and do they have an agenda such as selling you something. Many owners have bought tons of items for their dog that they do not need.