When most people think of Hinduism, what comes to mind is the myriad of Gods and Goddesses (I'll explain why I capitalized them later) that are mentioned and known.
Gods and Goddesses were in prominence primarily in the period of Vedic religion. The idea here was of sacrificing to the deities in order to gain worldly boons and merit for the period of time in-between lives. If you look at Mesopotamian religions, early Judaism and the Greco-Roman religions, you'll see that the sacrifice to God(s) plays a large role in worship. In addition, you also see other practices that are shared, such as purification/cleansing rituals, magic, and divination.
As time goes on, you see that many of these practices fall out of use for one reason or another, and this is especially true of polytheism as a whole, and more specifically animal sacrifice. In India, you have a rise in Puranic religious traditions which manifested as a rise in a trinity. This, depending on your sect, held either Brahma or Shakti/Devi as Creator. Vishnu was primarily seen as the preserver and Shiva as the destroyer. Various sects arose and one of these four deities was chosen as their center of focus, the most important out of the trinity.
In practice, most Hindus today have practices that are based on the sects that arose from Puranic tradition, and only truly worship one deity. All others are seen as "lesser" manifestations of that one Supreme God(dess). As such, you have the ideas that all Gods and Goddesses are legitimate in their own right even though they may not be considered to be the "highest" form of divinity. (Hence my capitalization of "Gods.")
However, there is also another major ideology of the idea of "God." This concretely begins in the Upanishads (उपनिषद्-s). The traditional "books" of Hinduism are the Vedas, and these have been passed down orally for a VERY long time. The samhita-s (संहिता-s) are the original hymns to the deities, and are considered the first of four parts. The fourth section of the Vedas is the Upanishad texts. These are more conceptual in nature.
There are a few general themes that appear, and major concepts develop: the older idea of time as cyclical is reinforced; the other planes of existences (Heavens, etc) are explained; the practice of asceticism and meditation is developed in detail; and, a monist view of the Supreme Godhead is developed. This Supreme Godhead is referred to as "brahman" (IAST: brahman, devanaagarii: ब्रह्मन्, not to be confused with IAST: brahmaa, devanaagarii: ब्रह्मा, the Creator God). This Godhead encompasses all of creation, but is beyond just creation. Some people have told me that they believe in "the powers of the Universe." I think that that phrase defines brahman pretty well.
So now, you have this brahman entity that essentially at the top of the list of Divinity. This gets identified later with devotional Gods such as Vishnu and Shiva. Thus, by worshipping Vishnu, or His incarnations (Rama or Krishna, for example), one is directly worshipping this personal idea of God, as well as the underlying divinity of the Universe.
Using that kind of logic, then why can't the older Vedic Deities also be worshipped? Let's take the Adityas (आदित्य-s) as an example. The Adityas are a group of solar Deities, of which Vishnu was originally a member. Surya (सूर्य) is considered the manifestion of the Sun. He is often identified with SavitR (सवितृ), from the Gayatri Mantra (तत् सवितुर् वरेण्यम्), and with HiraNyagarbha (हिरण्यगर्भ) (via the Sun Salutation). HiraNyagarbha means "golden womb," and appears in the HiraNyagarbha Sukta as the source of the manifested/created Universe. If I choose to worship the sun, or the Adityas in general, then in turn, I am also worshipping the Universe and brahman which is underlying it.
Let us keep in mind that even scholar disagree on many points, including: which Adityas are referenced in the Vedas and which are identified later; and which Deities separate and which are identified by multiple names (all referring to the same singular Deity). Thus, it becomes difficult to tell which God is referenced where, and whether that God is another by a different name or a separate deity entirely. My example above is simply my interpretation based on what I've learned. Feel free to do your own research and form your own opinion, and don't be put off by the fact that it's not concrete. That only adds to the fun.
This allows us, as educated Hindus, to choose what manifestation of God we want to worship. Here, I am reminded of a famous metaphor. "Just as rain falls and goes to the ocean, so do all prayers to all Gods go to the Supreme." As a Hindu, this is a really important source for religious acceptance of all others.
Perhaps more importantly, we gain the concept of monism. We can interpret anything and everything as a manifestation of God and beautiful in its own way, and thus we can better exemplify the message in the following verse:
यो माम्पश्यति सर्वत्र सर्वंच मयि पश्यति |
तस्याहं न प्रणश्यामि स च मे न प्रणश्यति ||
(भगवद्गीता ६ - ३०)
He who sees Me everywhere and sees Me in everything,
To him I do not perish and he does not perish to Me.
(Bhagavad Gita 6:30)
This is my own translation; most others translate न प्रणश्यति as "is not forgotten," "is not lost," or even as "dwells in." I chose to stick to a stricter translation, hence "does not perish."
To me, I see this as a practical reminder of the Golden Rule, but taken to an extreme. Instead of just doing to others as you'd have done to you, you are remembering the inherent monism present in all. We all have a spark of the divine in us, and it helps to have a reminder in some of our more difficult situations.
Just as we can choose which deity to worship, I don't see why we can't skip the idea of manifest Gods altogether. I'll eventually outline the benefits of a "personal deity" in a later post, but right now I'd like to point out that many are perfectly fine with and have no trouble paying homage and believing in an abstract, omniscient and omnipotent God. I see no harm in skipping the "middle man" so-to-speak, and going straight for the underlying source. This is not to say that polytheism or panentheism is inferior in some way. Both panentheism and monism have their pros and cons. I'm just saying that having room for monotheism and monism would allow more people to take religion into their own hands, and have it become more accessible to them.