Are blood tests, echocardiograms, EKGs, and X-rays used in combination, good ways to spot heart issues such as angina and artery problems?

Not necessarily. None of the tests will rule out heart disease with 100% certainty. Your effort should be to have a healthy diet, maintain healthy weight and exercise at least 150 minutes/week, to prevent heart disease.
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To diagnose angina. To diagnose angina and coronary artery blockages a careful history of the patients symptoms is important and, if suggestive, treadmill stress testing with or without nuclear study is performed. If results are abnormal, invasive testing by coronary angiography is the gold standard for documenting presence of significant coronary artery disease. Coronary artery calcification score isn't enough.