Native speakers recorded at forvo.com are using the "a" sound, for what it's worth: https://forvo.com/word/%D0%BF%D0%BE%D1%8D%D1%82/#ru
"Е" is usualy pronounced like "ye" in "yes" if preceded by the vowel or like "e" in "men" if preceded by the consonant. "Э" is like "а" in "cat". While there are instances where "е" sounds closer to "э" (like "интернет", компьютер), they are different letters, so I'm not sure I can give you a simple guide where you "should use which". I guess the only coherent answer would be "when the spelling of the word requires it". I can only note that "э" almost never follows a consonant (however there are exceptions to that too, like "сэр", "мэр", "сэкономить", etc.).
Э is pronounced like the "e" in "end", or like "eh". It sounds nothing like the "a" in "cat". Here's a pronunciation guide for the Russian alphabet: http://www.russianforeveryone.com/RufeA/Lessons/Introduction/Alphabet/Alphabet.htm
As a native Russian speaker I'd like to note that most native Russians won't hear the difference between "e" in "end" and "a" in "cat". so it's hard to say whether Russian "э" sounds "anything like it" :) In fact it takes quite a lot of training for a Russian to be able to tell the difference between "men" and "man". All those sounds are just slightly softer "э" for us at first.
This is such an interesting topic, thanks for your comment. The Wikipedia page suggests that Russian does make a distinction between the sounds [ɛ] (e.g. the e in men), as the usual pronunciation of э for example, and [æ] (the a in cat), which does come up in Russian as the sound of я between a consonant and another soft consonant, e.g. пять. When I raised this with another Russian native on a different page he said that Wikipedia was wrong. What are your thoughts?:
Well, from what I can tell, while the distinction exist in the language most native speakers don't register it. It happens subconsciously, and if one is not a linguist they might never realize they use different sounds. They just put it in the same mental "box" labeled "э-sound". In fact a non-native speaker whose own language "officially" makes such distinction would be more tuned to the differences. That's what happens when native English speakers hear the distinctions in Russian vowel sounds that Russian speakers never even thought of. Conversely, in my experience, most English speakers don't really think, for example, that they use different "l" sounds in different words, but Russians do hear the difference and some loan words or names are transliterated with "л" while others with "ль".
My impression so far is that э is also used so much less often compared to е. The only use cases of Э I know are это, эгоизм, and поэт (and their declensions); even though my vocabulary set is really small, there are only this few. I see you're a native speaker, could you verify this claim? Thank you!
That is correct. Щ, Ф, Э, Ъ are relatively rare compared to letters like А, И, Р, С, Е. Still, э is found in "это", which is an extremely common word— just like in English very few words start with a voiced th , except "the", "this", "that", 'there" etc (all very frequent).
The most common Russian words with an Э are это (and its detivatives), interjections эх, э-э-э and эй, words этаж "floor", экзамен, электричество, электронный, экран "screen", эксперт, эмоциональный, экскурсия, мэр "mayor", аэропорт, экономика, поэт, элемент, энергия, эпоха, эра, эффективный, элита, эпизод, экспорт, эксперимент.