Heart disease is a quiet killer. We see that in humans and the same occurs in our pets. It is perhaps even worse because our pets can't tell us that they're having a hard time going on that walk or fetching.
Dogs with heart disease will start to cough. Cats often don't. Juvenile pets born with heart disease may present differently. The most typical symptoms are coughing, lethargy, decreased appetite, darker gums/mucous membranes and an increased respiratory rate. Cats with heartworm disease may vomit frequently.
Your vet will want to see your pet if you call with any of these concerns. He or she will do a physical exam and recommend radiographs, bloodwork and possibly a cardiac ultrasound. If there is cardiac enlargement or fluid in the lungs, your vet will promptly start therapy. Your vet may tell you that there is a heart murmur, rhythm abnormality, rapid breathing or increased lung sounds. Some pets with heart disease may not have a murmur.
There are some valuable things you can watch for at home. If you think your pet has heart disease, please try to get a RRR (resting respiratory rate) before you come in. This video from a cardiologist goes over what that is and how to get it: http://veterinarymedicine.dvm360.com/how-owners-can-help-monitor-heart-disease-their-pets.
Keep in mind that although heart disease is quiet, there is help. Don't be afraid to pick up the phone and discuss your concerns with your vet.