My next project is to read the Bible, but I want to get the version that would be most representative/accepted by the widest swath of Christians possible.
The King James version seems to be one of the big ones and would be my guess, but the language seems a bit dated now.
I realize this is a judgment call, but if you had to make a pick, which would you choose, and why?
I’m assuming you want an English translation? If so, I think King James is going to be your best bet. I’m not sure why the older modern English use is a concern, but in my opinion and experience, KJV is certainly the version that most English-speakers are familiar with. It’s the source of most literary quotes of “The Bible”, and it’s one of the most pleasurable versions to read (King James put lots of poets on the case, including Ben Jonson).
You do know that a “Catholic” Bible has more books than a “Protestant” bible, right?
If this distinction interests you, you may be interested in buying a “Catholic” Bible or a “Protestant” Bible that includes what some people call the “Apocrypha” (though the word Apocrypha itself has different meanings depending on your denomination - a Roman Catholic and a Lutheran will not mean the same books when they speak of “Apocrypha”.)
King James Version containing the Apocrypha
From the multi-part Straight Dope staff report “Who wrote the Bible?” we learn
For Protestantism, the NIV. Because it fits your OP criteria.
I’ll second the NIV (New International Version), although I use the New Revised Standard Version, myself. If you like, Bible Gateway has some good information on various versions and you can compare versions side-by-side in it. I’ve been using it to back up arguments in debates about Christianity for years.
I’m reading the entire Bible this year for the second time.
This year I am reading the KJV. It is difficult to follow at times, but certainly isn’t impossible. However, if you’re not familiar with the story that you’re reading, I’d say KJV wouldn’t be a good choice. You’ll find yourself looking things up.
Most of the liberal churches I’ve been to use the New Revised Standard Version. This one is commonly used in college courses on the Bible. I own a copy and I find the notes and maps very useful. In addition, it has the Apocrypha so you can read those Catholic books! 
The NIV is probably the most widely used translation among conservative Protestant denominations.
A couple of other translations which are commonly used are the English Standard Version and the Revised Standard Versions.
I’d strongly suggest getting a study version of translation you decide on. For each of the common translations, there are study versions which have notes and maps.
The Eastern Orthodox Bible has even more OT books than the Catholic Bible.
For a standard English Bible, I’d recommend the Revised Standard Version (the “English Standard Version” is a slightly-to-the-right revision of the RSV).
After that, either the New King James or the New American Standard versions.
There are a lot of conservatives & liberals who regard the New International Version as too biased and more paraphrasal.
Except for some jitters about Isaiah 7:14 (“A young woman or a virgin shall concieve…”), many conservatives have reconciled themselves to the RSV, not so much to the gender-inclusive language New RSV.
The RSV & NRSV also come in Catholic & Catholic-Orthodox versions with the extra books. The standard Catholic translation to English now is, alas, the New American Bible. My preferred Catholic English translation is The Jerusalem Bible (or The New JB).
I would say it depends on what you are looking for.
In terms of its influence on the language of the West, the KJV is far and away the first choice. If you want an accurate translation, the Revised Standard Version is probably the best bet. The RSV and New RSV are copyrighted - many other translations are available here.
Regards,
Shodan
I agree that it’s very good reading, but I don’t think it’s correct that there were any poets as such involved, nor was Ben Jonson. All of the Translators were clergymen, except for one - Sir John Savile, a classical example of the learned Renaissance courtier, was the only one not in holy orders. (See God’s Secretaries by Nicolson, at p. 168.)
Of course, poetry was much more of a part of the culture back then, and highly learned men like the Translators would be trained in poetry as part of their education - some would likely even have written some, just like other educated men of the period - but they were selected for their knowledge of scripture and languages, not poetical abilities.
Nicolson points out that one of the reasons the KJV does have that appeal is that it was written for the purpose of being read aloud (“appointed to be read in churches”). The final editing process took place with one of the Translators reading aloud the proposed text of a particular book, and the other Translators listening. It wasn’t a written editing process, with mark-ups of the text being circulated for comment, but an oral process. If it didn’t sound right, then changes would be suggested. That emphasis on the spoken value of the text is one of the reasons, Nicolson argues, that it has had such long-lasting appeal, unlike texts of the same period such as the Geneva Bible, which was more popular at the time for private study, but has fallen by the wayside now.