I'm fixing a 1988ish Miyata 315 with a Tange ME S II headset. The headset is slightly stiff. When riding the bike, the steering doesn't naturally follow any adjustment to the arc of the turn without some extra rider input. If I hold the front wheel off the ground and turn the handlebars, the handlebars resist turning.

What's involved is adjusting and repairing the headset?

Remove the handlebars, unscrew the headset locknut and headset nut, drop the fork out of the frame, clean, inspect, lubricate the bearings and races, reassemble, and adjust.

"1988 Miyata"

You didn't mention a notchy feeling. My first thought was an upside down bearing but I can't imagine that being unnoticed for 25 years. My second thought was the natural tendency of a headset would be to loosen itself which obviously isn't the case either.

I'm thinking some kind of foreign material may have gotten into the bearings. I'm going to agree with RoadGuy. Disassemble, clean, inspect and reassemble the headset. That should do it and it's a relatively simple task.

If your headset does feel notchy when you turn the handlebar from left to right, your headset is hosed and needs to be replaced.

What tools are needed?

Most likely 2 headset wrenches for a threaded headset:

Park Tool Co. » ParkTool Blog » Threaded Headset Service

You can sometimes get away with Channel-Locks or very big adjustable wrench for the locknut, but need a headset wrench for the thin top race.

Does it look basically like this?

Yes it looks like the one in the picture. It's a 1 inch threaded with a quill stem.

I've been a non-professional shadetree bike mechanic for over 40 years, and I have headset wrenches sized between 30mm, and 36mm. And sometimes wrenching on an old bike SAE wrenches are necessary.

On a 1980s Miyata with a Tange headset, you're going to need metric wrenches. You may be able to use a regular wrench for the locknut, but you're probably going to need a headset wrench for the nut/adjustable upper race.

I don't recommend using pliers or vicegrips on headsets and bottom brackets, because you will slip, and you will damage the chrome, and that leads to rust. Get the right tools for the job and use them.

You can try measuring to see what size wrenches you need, but you're probably better off taking the bike down to your LBS to match the headset to the wrenches you see (unless you are willing and able to buy an assortment of headset wrenches for your toolbox). While you're at it, get the bottom bracket wrenches your bike needs. Sometimes wrenches have a headset wrench on one end and a bottom bracket tool on the other end. Having your bike at the LBS when you are choosing wrenches sometimes makes the decision about what wrenches to buy for you. You might have a choice of headset wrenches that have bottom bracket tools on the other ends, some of which will fit your bike, and some which won't.

1" threaded steerers often use headsets that require 32 mm wrenches. Headset wrenches are thinner than the usual "Crescent" wrenches because of the lower-profile locknuts as with the above photo. You would need two wrenches to adjust the bearing preload.

And once you have the proper tools, the easy part is taking it apart, cleaning, re-lubing and reassembling. The tough part is getting it properly adjusted. The trickiest adjustment on a bike, in my view. Takes patience and a bit of a touch to get them just right.

Job completed. The headset needed cleaning, lube and an adjustment. The steering now feels natural.

It could simply be dried grease. Headsets don't magically tighten, so the likeliest suspects are dry grease or rust.

Following the theory of trying the simplest remedy first, turn the bike over and apply a few drops of oil to the gap between the bearings. Swing the bars back and forth to draw it in, and continue to do that to circulate it and work it into the dried grease. If things begin to loosen and feel more free, we're on the right track, and a bit more time will soften the grease and restore to nearer to new condition.

Of course adjustment may be needed, but in almost all cases old bikes have headsets that are loose rather than tight.
Also know that the single most common cause of tight headsets (other than lube) is a fork that's bent back slightly, usually from a front end collision. So if it isn't dried grease or rust, sight down the handlebar stem, headtube and upper end of the fork blades like a ***** and verify that all are in line. (IMO, this is the first check to be done on any used bike).

BTW- if, as you ride, redish/brown oil or grease weeps out of the bearing, then there was some rust, but if it feels good and steers well you're good until you're ready to do a proper service on the headset.

Yes, I inspected the frame and fork before buying. No signs of damage. I've since had other experienced bike collectors look at it and they concure. The grease was factory installed and the overhaul, cleaning and regrease did the job.