I have about a 10 year old Fuji road bike. The bike is in great condition, very light, looks like new and only gently used. My question is, as I am about to start putting in a great deal more mileage, should I look at upgrading my bike (new gears, other componenets, etc. to go from a 16 to 24 gears, if this can even be done) or look at buying new?

TJMAX

It may need a few things to be done to it depending on how it was stored.

• If the tires are glassy feeling or have developed surface cracking (called "checking") then you'll want to replace the tires so you have decent grip and don't need to worry about a tire failure. Rubber tires age poorly and the rubber hardens and cracks in extreme conditions and age. At that point they are dangerous to use because they don't develop the same grip as new and the threads in the carcase are no longer supported well if the tread and sidewalls are cracked. Assuming you stored the bike in the garage where it likely gets stinking hot each summer it is likely that your tires will be shot. Either cracked or gone all "plastic'y" and be slippery as a result. Heat is the destroyer of rubber due to making the rubber age far faster than if kept in a cool dark place.
• Again, if it was stored in the garage where it got stinking hot in the summer then the brake pads may be prematurely aged and brittle depending on what material they are made from. Watch for signs of the pad surface crumbling during your first few miles or for signs that the pads are not stopping well. Replace if you notice either.
• The wheel hubs, and headset might do well to be stripped apart, cleaned and relubed after all this time. If stored in that same extra hot garage the grease can dry out over a decade to a rather crusty and non lubricating grunge. Try turning the steering slowly and feel for smoothness. Take the wheels off the bike and turn the axles by hand. If these bearings are running smooth then you don't need to do anything. If they seem a bit lumpy feeling then the grease has congealed into lumps and it would be good to clean and re-grease.
• IF the chain has links that are sticky try cleaning and oiling it. If it still has links that are sticky then get a new chain. Mind you in Arizona you're likely fine on this count and a cleaning and re-oil will have it good as new.
• Wash the poor thing. It's got a decade of dust and bug poo on it...
• Ride

And there is certainly nothing at all wrong with a 10 year old bike. Nothing at all has changed in any way to make your old ride obsolete. You'll easily get another 10 years of high milege out of it.

Ride it and replace the bits that wear out.
If you do decide to upgrade the gears, going to a triple with an 8-speed drive train probably won't be very cost effective. You could get a compact double crankset if you need lower gearing.

Is your present bike limiting you in some way? If not, or even not sure, save your money and keep riding what you have until you have a compelling reason to buy a new bike.

If you need a lower granny-gear, get another crankset to accomplish that, ..
Its not how many gears it's how the ratios that are available work for you

You can improve the ratio selection without increasing the number of cogs on the back wheel.

https://sheldonbrown.com/gears/

On the gear number count. There's no need at all to get all caught up in the "more is better" nonsense. You've likely got 7 or 8 speeds in the rear and two up front? That is more than enough to get out and ride your butt off.

IF by some chance you end up joining a riding group and find that you're looking for a speed inbetween due to their usual cruise speed you have a couple of options. First would be to change your cassete to one that has the inbetween gear you want. The other would be to THEN give in to the Dark Side and get a new bike that has that inbetween gear you want. But until you find a need to lust after this magical gear ratio there's still lots of great riding to be had with your Fuji.