An EKG gives a TON of information about your heart: Its size, its orientation (upright in chest, horizontal, on the wrong side of chest, which happens rarely), enlarged chamber, valves OK or no, electrical charge and discharge running right (this is where heart block shows uo really clearly), and so on. It is so quick that it looks like a simple little test, but it takes a lot of practice to read an EKG well, and the cardio who reads it for you or OKs the intern's interpretation is more skilled than you can believe.
Here is more info about what an EKG shows:
http://www.nhlbi.nih.gov/health//dci/Diseases/ekg/ekg_show.html
If you heart is performing normally when the EKG is done, the heart will appear normal. You said the doctor "saw" some minor arrhythmia, still that is almost normal. If you have other symptoms, including times when your are not on an EKG or being examined by the doctor, you should ask for a wearable monitor, perhaps a so-called Halter monitor which one wears for a 24 hour period would be enough. You'd be the one to know as if you have no symptoms while wearing, then it is unlikely that the monitor recorded any problems.
The next step, or a parallel step, is a stress test. One would go to an echocardiogram only if a physical/structural problem is suspected.
All the above are non-invasive, and thus can be done with minimum risk.