Sumerian is a language isolate. It has no geneological connection to any language that we know, in spite of the fact that there have been many attempts to connect it to everything under the sun. Yet, as you correctly point out, we find many words of Sumerian origin in Semitic languages. e2-gal ("great house" == palace/temple) appears all through Hebrew (היכל), Arabic (هيكل), or Syriac (ܗܝܟܠܐ).

To understand this we have to go to the 3rd century BCE. From at least 2500, and maybe earlier, the Sumerians lived side by side with a Semitic people whom we now call Akkadians, who spoke a language which we conveniently also call Akkadian. Their influence on one another was enormous, to the extent that scholars such as Jean Bottero refuse to speak of different Sumerian and Akkadian civilizations and prefer to speak instead of a Mesopotamian, bilingual, but unified culture. We don't need to agree or disagree with that, just as long as we understand the extent to which they impacted one another.

How are we to understand that? It's not so easy. Precisely because they were so connected, it's often difficult to pinpoint exactly what is Sumerian and what is Akkadian/Semitic. Often we come across something that we thought was Akkadian, only to find evidence for an earlier attestation in Sumerian. Or vice-versa, we believe something to be Sumerian, but we either find something earlier in Akkadian, or else are left in suspense by serious historiography that contends that the presence of Akkadians in the region of Mesopotamia goes very early, and so we can't really tell properly.

Religion is a particuarly difficult nut to crack. While we have a fair understanding of what the Sumerian Pantheon looked like as opposed to the Akkadian one - the earlier being headed by the three gods An, Enlil, and Enki, and the former by those three + the god Marduk, it turns out that this is a fairly late division in historical terms (with Marduk appearing only in the early 2nd millennium, so around 1900). Presumably the Akkadian Semites had a pantheon identical to that of the Sumerians, with their different names but not much else.

Now, for your question, language. Sumerian is, as I said, a language isolate, so we have no help from any other language if we want to learn it. The only outside source that we have for it are resources in Akkadian. After the 3rd millennium, and as the so-called Neo-Sumerian empire (2100-2000~ BCE) faded, Sumerian faded both as a language and as a distinguishable civilization (though there is a lot of controversy even on this!). Yet the language was retained as a language of culture for more than 2000 years afterwards, the latest texts coming from 100 CE (!). To learn it, Akkadian scribes devised a number of techniques: we have grammars of Sumerian written in Akkadian, we have word-lists giving the Akkadian and the Sumerian correspondent, and we even have interlinear Sumerian-Akkadian texts, probably of a scholastic provenance.

What this means is that most of the traffic of linguistic influence comes from Sumerian towards Akkadian. This must have happened in two stages: words that came into Akkadian while Sumerian was still a living language, and words that came into Sumerian /after/ Sumerian was only a "learned" language. Sometimes we even have the same word being imported into Akkadian while Sumerian was a living language, then Sumerian evolving and changing it, and then the word being importet again with a slighly different meaning.

This happened because, as mentioned, Sumerian cast a large shadow over Mesopotamia throughout the centuries. Even in the times of the Neo-Assyrian and Neo-Babylonian empires (900-500~ BCE) we see rulers and scribes adorning their texts with Sumerian words, not unlike the way it was not unfashionable to graft Latin words or expressions in speeches, or in scientific contexts, up to very recent years.

Influence of Akkadian into Sumerian is less explicit, but is nonetheless abundant. There's a fair share of words (like ragaba, from Akkadian rakibum, a perfectly sound Semitic root RKB still extant for example in Arabic ر ك ب). My favourite, and most intriguing, is the word u. In Semitic languages, as you know, و or ו or ܘ, the so-called "waw/vav" is the word that means "and". Sumerian, as far as we can tell, had no word for "and". They got by by juxtaposition, or through other methods. But very early on (I don't recall off the top of my head, but I think 2300~) we start seeing the word u in Sumerian texts. It's uncertain what this could prove. Some scholars suggest that this is a proof of non-native speakers writing in a language not their own, and finding something so inescapably alien as the absence of a copula word, that they add it in. Others, such as John Hayes, contend that this is precisely how things work: the first thing a language that does not have a word for "and" would do when living side-by-side with a language that has it would be precisely to import it! Be that as it may, the use of "u" for "and" in Sumerian texts increases exponentially after the decline of Sumerian as a living language: meaning from the moment that everyone writing in Sumerian was actually an Akkadian speaker.

So how did those words make it from Akkadian into other Semitic languages? It depends on the word, some must have crossed relatively early by the mid 2nd century BCE, while others had to wait until the later Neo-Assyrian and Babylonian empires. This period is, demographically, a big story of the ascent of the Arameans, which took over Mesopotamia and the Levant not politically but certainly linguistically. It would have taken words from Akkadian/Assyrian/Babylonian (different names depending on the historical context and the region, but just one language) and spread them throughout the land. The Hebrew Bible even contains books in Aramaic and plenty of other words the further in time you get. Apart from that, in spite of attemps by Greek rulers to hellenize the region, Greek remained a language of for the cities. When the Arab armies of the Rashidun conquered the region, the majority of the population spoke Syriac/Aramaic (again, same language depending on the context), and would have gotten the by now ancestral Sumerian words from them.

For sources dealing with loan-words from Sumerian into Akkadian or vice-versa, I'll venture to suggest you take a look at the 1st and 2nd chapters of Jean Bottero's "Mesopotamia: Writing, Reasoning, and the Gods" for a wider net, or else something like this for a more direct link and more immediate source.