What, then, are the mains tenets of Theosophy?

Consciousness, Universal and Individual

First, that the universe is not just a place where nature's forces operate by chance. Every event that has happened from the beginning of time has happened according to certain laws inherent in the universe. These laws are the expression of a Consciousness. Everything that exists, from the electron to the largest star, is impregnated with Consciousness. This Fundamental Reality is so far beyond our grasp that sages and saints have called it by contradictory terms. Many have termed it 'God;' but some have called it Law, Heaven, the Great Architect, Evolution. Each man, according to his temperament and his experience, must determine how he will regard this Consciousness which directs everything. Let us call it Logos.

Immortality of Man

The next great truth is that the nature of the Logos resides in every man and woman. We are not those bodies which perish; they are only garments which we wear for a while and cast aside. We are immortal souls. The perfection of Logos dwells in us also, for we live and move and have our being in Him. But we are unconscious of our Divine Nature, till we awaken to it.

Reincarnation

It is to realise our true nature that we are born. Our birth is as the entrance into a workshop or laboratory, where by work we slowly unfold our faculties. But it is not possible to rails the Divine Nature in us by the experiences of one lifetime. So we reincarnate again and again. We enter into life, we are born, we grow, we act, we finish our work and we return. Our return is death. After a rest in heaven, growing by realising the joys we planned but did not achieve, we are born again, more purified, stronger, wiser, to work again, so as to become more expert in thought and feeling and action. This is Reincarnation.

Karma

As we live and act, sometimes we succeed, sometimes we fail. We do good and we do evil, guided by our altruism or our selfishness. When we do evil, we create discord in the universal harmony, and we must restore that harmony. The evil we did must be balanced by new good; the good we add must be reshaped to a more far-reaching good. This process of sowing and reaping is called Karma. It is the law of readjustment, which a man puts into operation by every one of his thoughts, words and deeds.

ONLINE: Aspects of Divine Law - Geoffrey A. Farthing

Universal Brotherhood

Since all souls are divine, all souls are equal. There are young souls and old souls, but all are brothers. In spite of every difference (of birth, capacity, and environment; of race, creed, sex, caste or colour; of goodness or wickedness) all people form an indivisible brotherhood. All of us, high or low, ignorant or wise, make a chain, and the stronger grow by helping the weaker. Brotherhood is the law of growth for all persons.
But this Brotherhood extends to all (animals, birds, and fishes, even the plants, mountains and seas). We grow by our unity with all things. The Divine Nature, which is latent in them as in us, helps our inherent Divinity to step forth in its beauty.

God's Plan Which is Evolution

There are two eternal mysteries: the mystery of Logos and the mystery of Man. All that we postulate concerning Logos, of goodness, holiness, truth and beauty, resides in man. Religion, philosophy, science, the arts, commerce, industry, philanthropy, all these are channels along which the Divine Nature descends to reveal its Beauty. And our human nature ascends to Divinity by growing in the virtues and capacities associated with these channels.
Theosophy means the Wisdom of God. It is a statement of the modes of action of the Divine Mind. All our questions are solved when we understand 'God's Plan, which is Evolution.' This understanding is the heritage of every soul. But he will possess it only as he learns to be a brother to all that lives, for 'Loving action is the Divine Wisdom at work, and who so acts lovingly will inevitably come to the Wisdom.'
Members of the Theosophical Society study these truths, and Theosophists endeavour to live them. Everyone willing to study, to be tolerant, to aim high, and to work perseveringly, is welcomed as a member, and it rests with the member to become a true Theosophist.