Suggesting that the Sumerians originated from a single place is often a mark of pseudo-history in the field of Assyriology. For a more nuanced take on the so-called "Sumerian problem," I recommend chapters 3, 4, and 5 of Georges Roux's Ancient Iraq, "From Cave to Farm," "From Village to City," and "Birth of a Civilization," which explores the archaeological evidence from paleolithic settlements up through the Jemdat Nasr period, tracing various regional influences that likely fed into what would become the Sumerian civilization.
I recommend starting with Ancient Iraq by Georges Roux (1993). Roux covers the region from prehistory until ca. 200 BCE, discussing the accomplishments of the Sumerians, Akkadians, Assyrians, Babylonians, and various peoples who came to power in the region after them. From there, you can decide if you're most interested in the culture, mythology, politics, religion, etc., after which it will be easy to recommend further books focused on your specific interests.
If you already have a specific topic in mind, let me know and I'll see if I can suggest a good book or two focused on it.
While our community is dedicated to "Mesopotamian religion," its important to remember that there are actually four religions included under that grouping:
Many aspects are shared across all four, like the names or general domains of certain deities; but there are also practices and beliefs that are unique to one or two and not the rest.
When it comes to "getting started," I always recommend a general history of the region as a precursor to practicing any of the religions because learning about the four major kingdoms will help you decide which of their religions you're most interested in practicing.
Currently, my favorite overview of Mesopotamia is Georges Roux's Ancient Iraq, available in its third edition from Amazon.
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If you already know about Mesopotamia, or have a specific deity (or deities) in mind, then the second step is to establish a general practice.
Most of us begin with a simple set-up: an offering dish and a cup for libations. These should be kept in a space set aside for spiritual activities: a designated countertop, a nightstand, an altar or shrine if you already have one, etc.
Once you know where you'll be keeping the offering dish and cup, the next step is to figure out a schedule you can adhere to. My advice would be to start small. If you can only provide an offering and libation once a week, then provide it once a week.
The intent and consistency of your actions is more important than their quantity.
When you know which day you'll be providing an offering and libation for the deity of your choice on, you can then decide what you would like to provide for them.
There aren't any restrictions to this step in the modern day, but I'll give you a general overview of what kinds of things were offered historically:
Use the above as a guide to creating your own offerings and libation. Again though, don't over extend yourself. If you can only manage a small offering, then provide a small offering.
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And that's where you begin: with a regular schedule of providing offerings and libations to a deity or deities of your choosing.
From there, you can expand the space, making it into a shrine with an offering table. If you find that you're interested in certain practices from Mesopotamia (magical or otherwise), you can add an altar to the set-up as well. Some of us have elaborate shrines with altars, others of us have very simple spaces with just the bare essentials for daily or weekly meals. The growth of your sacred space is entirely dependent on how invested you are.
Finally, if you discover a deity you really want to focus on, or if one responds well to your offerings and libations, then you can become a devotee. This is when you'd do things like seek out a physical representation of the deity (a "cultic statue" as its called); replicas of their symbols, weapons, and standards; mythology, cult songs, and paeans of praise honoring them; and the festivals and holidays that were dedicated to them.
Most people want to start with this final step, jumping immediately into devotional work and holidays. Often though, without the basic structure provided by the gift-giving cycle of offerings and libations, people burn themselves out wondering why they aren't getting a response when they celebrate a holiday.
In my opinion, you have to first show them that you're serious about practicing the faith, then you can celebrate with them.
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This is a very bare-bones overview. I'm sure you'll have questions, so feel free to ask them and the community will do its best to provide answers and guidance!
A classic from my undergraduate days. I assume it's dated.
https://www.amazon.com/Ancient-Iraq-Third-Penguin-History/dp/014012523X