Excellent question!

I understand William H. Nienhauser has been trying to translate these in full for a long time

Translating anything written in classical literary Chinese is already tremendously challenging in itself, and any serious attempt at translating Sima Qian's one-hundred-thirty-volume historical tome is a monumental challenge. William H. Nienhauser over the years has done a tremendous job in terms putting together excellent multi-national talents, as well as towards securing long-term funding for his Shiji translation project. In this regard, The Grand Scribe's Records series by William H. Nienhauser and his team is by far the best scholarly English translation of Sima Qian's Shiji I have encountered so far.

By best "scholarly" translation, I mean Nienhauser approached Shiji in a way that is similar to how classical historical scholars in China did -- heavily annotated, with rigorous, almost "clinical" close-textual attention, and contextualizing the passages with other relevant classical Chinese texts. In this sense, Nienhauser's version can be seen not merely as a translation of Sima Qian's writings, but a fully developed, methodical annotated study of Shiji, done in a way that largely conforms to traditional Chinese historiography.

That being said, being the best scholarly English translation available does not automatically imply that the Nienhauser's version is the best translation for all purposes. The issue is a bit more complicated when considering OP's following comment:

but where should I start as someone with little to no understanding of ancient Chinese history?

And by "someone with little to no understanding of ancient Chinese history," I will also assume that the said reader is also someone without a lot of prior knowledge on ancient Chinese classics (i.e. The Analects, Book of Documents) and historical canon (i.e. Zuo Zhuan 春秋左傳 and Gongyang Zhuan 公羊傳), which would make Nienhauser's version somewhat inaccessible. Now it's a good time to turn our attention to another translator named by OP:

Burton Watson has also translated many parts of this text.

Burton Watson (whom recently passed away) is an excellent scholar of ancient East Asian literature, and I am very impressed with his stylistic clarity and general-audience accessibility in his translation works, Shiji included. BUT, looking narrowly from a scholarly angle, the Watson translation of Shiji, while beatifically done in its own right, is my least preferred version. The reason is two fold. First, rather than a strict "translation" of classical Chinese text, the textual style of Burton Watson's Records of the Grand Historian series could be more accurately described as a Westerner's re-telling of Sima Qian's work in contemporary English. Watson took a lot of literary liberty to make his "translation" highly accessible to the senses and sensibilities of general English-speaking readership, and for that I applaud his craftmenship. But Sima Qian's distinct prose style is completely erased in Watson's artistic "re-telling." This is a non-trivial erasure IMO, as Sima Qian's rhetorical style have fundamentally influenced the narrative structure of official historical records (or *Zhengshi / * 正史) throughout imperial China.

Aside from sweeping stylistic Anglicization, the Watson version also made significant revisions in terms of word choice, often replacing historically particular terms and concepts from the original text with more general and/or "modern-sounding" phrases in the English translation. It is easier to demonstrate what I mean here by looking at a concrete example, let us focusing on the opening line of Shiji vol.5: The Basic Annals of Qin 《秦本紀》 and compare the original text with Watson's translation of that line:

秦之先 帝顓頊之苗裔 孫曰女修 女修織玄鳥隕卵 女修吞之 生子大業

Let's first break down the literal translation of the original text, part by part: “秦之先 Ancestral origin of Qin 、 帝顓頊之苗裔 descended from the line of Di Zhuan Xu 、 孫曰女修 it is said one grandchild [of Zhuan Xu], Nüxiu。 女修織玄鳥隕卵 As Nüxiu weave, the Xuanniao fell its egg、 女修吞之 Nüxiu swallowed [the egg]、 生子大業 [and then] gave birth to a son, Daye.

Now let's look at the same line in Watson's translation, from his Records of the Grand Historian (Columbia University Press, 1993):

The forbears of the Qin were descendants of Emperor Zhuan Xu. Zhuan Xu’s granddaughter was named Nüxiu. When Nüxiu was weaving, a swallow happened to drop an egg. Nüxiu swallowed the egg and gave birth to a son, Daye.

Note that I have highlighted a few key terms in bold: First, the 帝/Di in 帝顓頊/Di Zhuan Xu is the title of the five legendary virtuous rulers (Wǔdì) of pre-Dynastic quasi-mythological period. Sima Qian himself explicitly stated in the postscript of Shiji (volume 135) that: "scholars generally agree that records concerning the Wǔdì are divinely inspired (学者多称五帝，尚矣)." In Watson's translation, however, Zhuan Xu was erroneously and anachronistically labeled an "Emperor." This is perhaps a practical compromise to avoid using terms that are unfamiliar to the general readership, but such "audience adaptation" strategy can be controversial...

Next, the original text simply labeled the legendary Qin progenitor Nüxiu as a "grandchild" of Di Zhuan Xu. At no point did Sima Qian ever mention Nüxiu being female, and there is no reason to assume so -- in Chinese classical text, it is an established norm to portray legendary ancestral figures (such as Nǔwa) from quasi-mythological era being gender-ambiguous or gender-less. And I must say similar appropriation of gendered labels there quite common throughout Watson's translation of Shiji.

Now let's move onto that "bird" who dropped the magical "egg" -- the original text identified the bird as Xuanniao 玄鳥 (lit. "Blackbird"), and those of you who are familiar with ancient Chinese mythology will immediate recognize its name. Indeed, it is that extremely well-known mythical beast which symbolizes human fertility, and appears in numerous Chinese classical texts, always magically appears to drop its egg and give divine conception of a legendary ancestral figure. In Shijing or Book of Odes, this Xuanniao 玄鳥 was regularly invoked as a direct euphemism for conception, as seen in the line giving blessing to new pregnancy: "簡狄在台嚳何宜 玄鳥致貽女何喜. In Watson's translation, however, this mythical Xuanniao 玄鳥 creature simply became "a swallow happened to drop an egg."

Now please forgive me, but this response is getting quite long, and I haven't even gotten into Herbert J. Allen's translation yet. I must take a short break and write a follow up response tomorrow to wrap up and tie together the remaining loose ends. So stay tuned!