REIMS--pronunciation ??

Rans, but gargle the r in the back of your throat.

You have picked a hard one for a number of reasons. As you have discovered, the pronunciation does not look anything like you would expect as an English speaker. I find the French ‘r’ very difficult to pronounce, esp at the beginning of the word. As bets says, you sort of gargle it out of the back if your throat. I sort of prolong the vowel, followed by ‘ns’ sound.

Don’t worry too much about it. You can always write it down if you can’t make yourself understood.

Yup, not like it looks.
Somewhere, possibly a RS guidebook, the suggestion is to say it like "France" without the F.

Reims is pronounced /ʁɛ̃s/ in French.

But OK, what does that mean? There are three sounds:

• /ʁ/ is the famous "French R", pronounced in the back of your throat.
• /ɛ̃/ is one of the 4 (or 3, depending on what variety of French you speak) nasal vowels. The spelling is unintuitive to an English speaker but generally conforms to the rules of French orthography†, where a final -ein/-eim or an ein/eim that appear before consonants produce this vowel. This is the same nasal vowel as in plein, peins, sein, saint, faim.
  • It's not the easiest thing to describe how to pronounce a nasal vowel in text, but basically you lower the soft palate at the back of your mouth as if you're about to say "[r]ang" (in, say, a General American accent) but your palate doesn't actually touch the tongue.
• /s/ is the final consonant and same as the English s in sit, sat, pass, etc.

†Yes, there are rules. There are numerous exceptions and idiosyncratic cases, the rules are complicated, and it's nowhere near as simple as Spanish or Finnish, but predicting the correct pronunciation of a French word from spelling is less random than people like to make out. (In particular the rules for reading vowels, while complex, are generally consistent. Final consonants are one of the areas that are substantially less regular.) It's the other direction -- predicting the correct spelling of a French word from its pronunciation -- that is much more complicated.

Watch the beginning of this video in French, at 00:16 Reims is the first word spoken.
The video talks about the history of Reims.
"Reims" is pronounced several times in the video. Another place where it is easy to hear is at 03:57

https://www.youtube.com/watch?v=-_bpLl-CXHc

When you have managed to pronounce Reims well, I suggest you try this name of this other city located near the Dordogne:

Castelnau Montratier-Sainte Alauzie

it sounds like clearing through throat -- sort of a gravely 'rance' ---

JoLui, that's just mean.

Go to forvo.com and you can hear Reims pronounced by 14 French people and one Belgian. You can hear subtle differences but this will get you in the ballpark.

"Rance" is what the anglo ear will hear

I'll second Crumbs' suggestion of forvo.com. It's a helpful resource to get a sense of how to pronounce unfamiliar words or place names. With French words, just be sure to favor people from France rather than Belgian, Swiss, or Canadian contributors. Those last three aren't necessarily wrong, but they commonly differ from people who live in France.

And place names can be quite interesting. Reims is one example. Paris is another (though the French pronunciation is widely known).

One is a place in southern Bourgogne near where I used to live: La Clayette: a pretty little town with a château next to a large moat/reservoir. It didn't take long to learn that locals don't pronounce it as Law Klay Yet.

Nope. Locals pronounce it "Law Klett." It's as if the "a" and the "y" weren't there.