Greetings of Peace and thank you for taking the time to introduce yourself to the forum. There have been a number of people participating on our forums over the years coming from a situation similar to yours. Some of them, by the grace of God and the support of fellow seekers, have been able to enter into a living orthodox Tradition and secure peace of mind by doing so. I am of the opinion that other readers may also benefit from participating vicariously in such a heartfelt personal quest as well as by sharing the personal testimonials of their respective journeys.
You may find Marco Pallis' Some Thoughts on Soliciting and Imparting Spiritual Counsel very helpful at the outset of your inquiry, answering some of the basic questions that you may have, while anticipating others.
Perhaps one of the most useful preliminary observations from that article for someone looking to enter a tradition is the realistic assessment that there are primarily three traditions that are most accessible to the contemporary westerner: Buddhism, Christianity, and Islam.
The short answer is that within most Traditions, there is no rigid and exclusive demarcation between the exoteric and esoteric domains, even in the case of organized sufism. The foundations of doctrine, practice, and virtue issue from the same source of revelation such that the difference between the two domains lies primarily in the degree of interiority and intellectual and existential participation. To attempt to pursue "esoteric affiliation" outside the context of the formal aspects of the Revelation is to make the error of the occultists. Typically what happens is that they end up trying to create their own personal religious and doctrinal systems while experimenting haphazardly with various forms of meditation and invocation. "It is not piety that you should come to houses by their rear," states the Quran, "but piety is he who is reverent and comes into houses by their doors. So reverence God, that haply you may prosper." (2:189)
This is a very common argument against the adherence to religion that I have encountered numerous times in my correspondence and most recently in my discussions with non-believers at my place of business. It is rooted in the appraisal of religion based upon the actions and teachings of those who are least qualified to represent it (i.e. those who re-interpret traditional doctrines to reflect their adherence to modern ideologies). A greater amount of thought on the matter quickly reveals the fact that such an appraisal is not based upon the consideration of the religious tradition itself (including the revelation from which it emerges, the pattern of human perfection demonstrated in the being of its revealer, and canonical rites and means of worship) nor upon those who are most qualified to represent it such as the great prophets, sages, and saints of history, or their living counterparts today. It may be that a given sermon, homily, or khutbah may be displeasing to one's own discernment, but it is very disempowering to close oneself off to the blessings of heaven due to another person's individual limitations and prejudices.
I will try my best to tread lightly on these topics in light of your firmly held beliefs as I do not wish to be offensive or presume that anyone (yourself included) must hold to the same opinions that I do. Perhaps we can simply view the exchange as a shared inquiry. For myself, I share a similar enthusiasm for these subjects, and believe that there are few if any other topics of discussion as important as they are.
To back up a bit and address some of the personal questions that you have previously directed toward me, in my late teens and early twenties, as a result of my extensive study of occult literature, I had become convinced that initiation through affiliation with secret or semi-secret esoteric organizations was the goal of the spiritual life and the natural aspiration of any person of reasonable intelligence. I affiliated myself with a number of organizations at that time -Theosophist, Rosicrucian, Hermetic, Qabalist, Quasi-Masonic, etc and found in each instance a certain degree of elitism. Their initiates were passed through elaborate hierarchical ceremonies indicating the stages of their spiritual advancement and they were introduced piecemeal to various collections of occult doctrine accordingly. They were taught to control the spirits of the elements, the intelligences of the planets, and participate as co-creators with the gods by invoking them and assuming their imaginal forms. Each organization viewed itself as a repository of ancient wisdom, of "things kept secret from the foundation of the world" and looked upon religious believers as passive servants to a corrupt ecclesiastical authority. The crowning achievement of the highest ranking initiates was typically a vision or communication written down as a document or book and viewed as a kind of personal revelation that may subsequently be shared with the organization and interpreted according to the prevailing hermeneutical methodology, typically mathematical and astrological.
Now, most people who adhere to these organizations believe firmly in both their authenticity as well as the privileged status that is afforded to them by virtue of their initiation. It was common for people to be initiated into multiple organizations, but for the most part the path of initiation was viewed as a closed system and the doctrines espoused by the organization as absolute truth.
Being the inquisitive person that I am, I continued to investigate outside of these systems and encountered among other things the original sources from which they borrowed the symbols, doctrines, rituals, and techniques. For the most part these consisted of orthodox and traditional doctrines, rites, and methods removed from their original context and combined together according to fabricated rules and reinterpreted according to popular ideologies.
My original investigations led me to the conclusion that two things were absent from these organizations, God and deified men. There was a strong preoccupation with one's own spiritual advancement that overshadowed the consideration of humble submission to God's Will. Also, although initiates were unanimously preoccupied with mastering and controlling cosmic forces and spiritual intelligences, there was no comparable indication of self-control or self-mastery. In short, there were no saints of occultism.
Later, as I became more focused on orthodox religion and like many others, the choice of religion, came the consideration of the centrality of Revelation. According to my present understanding, Revelation (what you referred to as the source of power in religion) is an eruption of the Divine within the domain of the human. It is when God, in his quality of Infinitude, limits Himself by communicating with humanity in a way that is perceptible to ordinary human experience and awareness. When such a tremendous phenomenon occurs, it changes the order and organization of the entire world regardless of the civilization within which it originally occurred.
The late Martin Lings - May God be Pleased with Him - communicated what I believe is one of the most beautiful and instructive metaphors in illustration of this phenomenon which I originally encountered at that juncture. He wrote,
"From time to time a Revelation 'flows' like a great tidal wave from the Ocean of Infinitude to the shores of our finite world."
According to Lings, the formal religion consists of that portion of the wave which conforms to the particular ethnic receptivities and aptitudes of that sector of humanity within which it was revealed and as such is the exclusive province of the vast majority of the faithful.
Likewise, Esoterism, according to Lings, "is the vocation and the discipline and the science of plunging into the ebb of one of these waves and being drawn back with it to its Eternal and Infinite Source."
Although the vocation or calling of the esoterist is to the ebbing wave, Lings also explains that part of the discipline of the esoterist consists in conforming precisely to the formal elements of the aforementioned receptivities and aptitudes for the following reason. Although the center of consciousness - the heart or Supreme Self - has the capability of being set free and ebbing with the wave, the body and soul of man require a vehicle furnished by the religious form to carry them through human existence to the shores of death.
As far as my own path is concerned, I studied many ways both orthodox and heterodox before encountering teachers such as Lings from whom I originally learned about and came to identify with the religion of Islam and its spiritual heritage of Sufism.
There are a number of orthodox turuq within the Islamic Tradition, each of which represents a unique but similar branch of the same initiatic chain reaching back to the Prophet Muhammad - May Peace and Blessings be Upon Him. It would be advantageous to continue with the theoretical activity at least briefly at the outset in order to acquaint yourself with the basic perspectives and practices of Sufism and identify some of the different turuq. There are a number of good books and some websites which can assist with this activity and I can suggest some if you are interested.
Practically, you may then wish to determine which turuq have a center in your area and make contact with one of them. This can be as simple as sending an e-mail or calling to speak with a representative. Some of the larger turuq, such as the Naqshbandi and Mevlevi, have a center in many if not most major metropolitan areas and include some activities that are open to the public. If there is a specific tariqah that you have become interested in but are unsure of whether or not they are represented in your area, you can also inquire about local activity and representation.
Most participants at a local mosque, Sunni or Shia (at least within my experience within the United States) are not likely to be able assist you with this inquiry. For the vast majority of Muslims here, Islamic spirituality is encapsulated within the canonical prayer, personal petitionary prayers, and the concrete moral teachings of the Quran and Sunnah.
I am glad that you brought this up as it touches upon an important point that I intimated previously, specifically that in practice there is not as rigid a demarcation between the exoteric and esoteric domains as there is in within the theoretical exposition of these concepts. Comments such as the following made by Guenon in Perspectives on Initiation may easily be lost sight of in the grander scope of his teachings. He wrote, "Now, for the sake of convenience we could divide traditional organizations into the 'exoteric' and the 'esoteric', although these two terms understood in their most precise sense cannot be applied with equal exactitude." (emphasis mine) Elsewhere he states in the same book that "... esoterism has more direct links with religion than with anything else in the exterior order by reason of the traditional character common to both [i.e. their emergence from Revelation]; and as noted previously, esoterism can in certain cases even assume a base and support in a specific religious form." In my experience, this latter situation is virtually always the case, aside from certain notable exceptions such as the Vedanta which in any case contains a movement away from exoteric rites rather than an a apriori absence of an exoteric framework.
Although the Islamic Tradition contains perhaps the clearest demarcation of these domains, Islamic esoterism also functions within the framework of exoteric Islam and possesses as its foundation the symbols furnished by the revelation and the law and rites contained within the fundamental pillars of the religion. To this are complemented certain supererogatory rites and a metaphysical doctrine which is the inward analogue to the aforementioned symbolism.
To respond to your friend's information, Sufism may be seen as a complementary path practiced within the foundation of the rites that he mentioned but which has at its heart an interiorizing grace transmitted from the origins of the revelation and supported by traditional esoteric rites and doctrines. You may wish to read through Guenon's article on Haqiqa and Sharia in Islam for a better explanation of what this entails.