Excellent. I can offer some more tailored recommendations with your specific camera in mind. Fuji's manual for this camera covers three different series; given that yours is the "highest" model number in the range, I'm inclined to suspect your camera will have all of the settings mentioned. If there's any question regarding accessing these settings, of course just ask.
You've a Full Manual Mode available, which is a boon. I'm guessing you're already using it. I'll go through various settings in the order in which they're displayed in your manual, on Page 70:
Scene Position: Having chosen "M," you won't be using this.
ISO: You apparently have ISO64 available. That's a very low setting - most cameras stop at 80 - and I have to wonder if they're doing something "artificial" to get there. One goal is to absolutely minimize in-camera processing of your images; I'm going to suggest ISO100 as a start point and 200 should be OK if you need it to shorten exposure times.
Image Size: L(4:3). Chosen only because the resulting image is as close to square as possible, meaning you're going to get the most-possible vertical pixels and therefore the largest final coin size in the sensor.
Image Quality: Fine. Use every bit of the sensor's capabilities.
Finepix Color: Standard. Refer to the above "as little processing as possible."
White Balance: Here's where you start playing a bit. White Balance is the camera's adjustment to the color cast of your lighting (color temperature) and I think you're currently using it on Auto. That's OK, as your colors seem right, but I advise switching to a monochromatic background - black, white or grey - to make the camera's task all the easier. You have the ability to do Custom white balance. Your camera wants to use white as the "base" for this measurement; use a small stack of printer paper as the "white" for this setting if you wish to utilize the capability.
Sharpness: Standard.
Face Detection: Off. Yes, most coins have a face but that's not what they're talking about.

Photometry: Ugh. Why don't they just call this "metering" like everyone else? Use "Spot" so the camera takes all metering information from the center of the image, i.e., only the coin.
AF Mode: Center, for the same reason as above.
The other settings as described on those pages are irrelevant. However, you might want to play with bracketing - this is where you shoot a series of 3 images for each shutter press, each varying in brightness. It's a good way to avoid multiple shooting tries when you're unsure of exposure, although experience will teach you how the image in your viewfinder relates to what you'll see on the screen and eventually you'll probably nail the exposure first-time, every-time. If you use this setting, use 1/3EV.
OK. Experience with varying P&S-type cameras (yours is one, the alternative being interchangeable-lens types) makes me think you're going to find your camera's aperture "sweet spot" at about f/5.6. Maybe one stop lower or higher. Use the widest (lowest numerically) aperture which gives you a satisfactory depth of field; P&S cameras can do this with wider apertures than a dSLR. A wider aperture gives you more latitude for lighting and exposure, always a good thing.
Your camera has a self-timer; 2 seconds is enough.
You are working towards aperture and ISO settings which allow you to set an exposure of 1/100 or faster, roughly speaking. If ISO200 and f/5.6 don't allow you 1/100 or faster for a lustrous coin (for that, I'd hope for 1/200 or faster as duller coins will require longer exposures), you need more lighting. Longer exposures increase the chance of outside vibrations affecting the shot. This may seem nitpicky to some, but it's a
definite factor in macro coin photography. You're pushing limits here which aren't normally pushed in photography.
That, in combination with my previous post, is a start. There are more refinements possible, but that falls into the category of "extreme tweaking" and we may not need to go there. This should give you enough to play with.
