I'm wondering is there any other Phonetic notation other than IPA — that is easy to understand by Native English speakers
There is a list of notation systems in the Wikipedia article Phonetic transcription. Whether any of them are easier to understand for English readers, I don't know.
But any system based on English orthography is likely to have at least three problems:
- English spelling is notoriously inconsistent in its representation of sounds.
- English spelling is realised very differently by speakers of different varieties of English
- English does not contain all the sounds, or make all the distinctions, that some other languages make, so if the system is to be used for languages other than English, it is going to be deficient.
IPA seems strange to English readers precisely because it is aiming to represent sounds, and make distinctions, which might not exist in English, and to represent them in a consistent way, which English orthography does not.
It depends on what you mean by "understand". I suppose you mean "learn to use correctly, to represent any spoken word" (of any language / of English). The biggest obstacle is making the correct association between the written notation and a pronunciation. English dictionaries typically give a list of symbols and example words, e.g. "a as in sat", "e as in set", "ā as in sate","ä as in sot", "o as in sought". This fails miserably if you don't how they think these words should be pronounced (sot and sought are just different spellings for the same pronunciation, as far as I am concerned). What you need is a standard set of pronunciations – recordings – that anybody can consult. This exists for IPA, and not for other systems.
If you do a bit of archeological work on the internet, you may be able to find pronunciation examples for the APA, which was the dominant linguistic transcriptional system outside of phonetics for a long time (used by field workers, anthropologists and language-area specialists). However, it is pretty easy to learn the articulatory descriptions of the various symbols, so if knowing that [ü] is a "high front rounded vowel" is a sufficient level of understanding for your purposes, then it can be understood.
If you work out the equivalences between APA and IPA, you can listen to IPA [y] and translate that into APA [ü], which would allow you to go from spelling to pronunciation in any language, not just one language and only in the dialect that you speak. The intermediate lingua franca for translating between the systems is the articulatory description. I don't think that one system has an advantage over the other based on using words like "stop" versus "plosive". I do think IPA misses the boat by not having a category "affricate".
However, your interest in "easy for English speakers" suggests that you have in mind "easy to figure out the pronunciation of a word in English", specifically, in your dialect". Imagine there was a word pronounced like super, only with f instead of s. The first vowel is actually pronounced many different ways, maybe [ʉ, ɪw, u] and myriad other possibilities. Typical non-phonetic dictionary systems will pick a standard way to write that vowel, maybe o͞o, oo, ü, ū, which doesn't tell you how the word is pronounced, it tells you "which vowel" it is, out of the set of English vowels. Then the vowel of "foot" will be dictionary-spelled with o͝o, u̇, ŭ, u or oo – of course, "oo" is not the foot-vowel in the same dictionary where "oo" is the "super" vowel. IMO, it is easiest for non-linguist English speakers to understand a respelling system that makes the fewest changes to actual spelling.
From the explication of the question I guess it is looking for some kind of popular phonetics like this example. They exist for a reason, namely explaining the pronunciation of foreign words and names to native speakers unaware of the foreign language. They are restricted in the sense that they only cover a few selected foreign languages (typically French, Spanish, and German) and that they offer only a limited accuracy in rendering the foreign words (so you still have an accent according to your native variety of English). Those limitations pretty well explain why they aren't used in linguistics at all.