Is it "Baby T-Rex?" (A red one) I am currently running a Baby T-Rex, and I can give you all the tips concerning that model that I know.
First of all, that "sad face" could be a DISCIPLINE problem. If the baby t-rex is SICK, a little thing looking like a mushroom appears in one upper corner of the screen... But if the t-rex is undisciplined, that "face" thing appears.
The advice: When the black face appears, your pet is VERY LOW on discipline! Discipline him/her once or twice. Be careful how quickly you discipline your pet more than once in a row, though -- if you discipline the baby t-rex too often, it will lose health! Once or twice should do the trick.
To increase the happiness bar, just play the game with him. (It's the icon with the guy playing basketball. When you start the game, a mouse or a truck appears; press the left or right arrow key repeatedly, depending on which side the mouse or truck appears on. You must chase the mouse or truck down to win!)
Increasing health is harder. There's no way to "manipulate" the health, as you can with some virtual pets. Health is gained only by taking good care of the pet. If you keep him well-fed and play with him, don't feed him treats too often and shower him at least once a day, his health will gradually increase. You can only take him to the doctor if he is badly sick (and the little "mushroom" appears).
Any other questions?