I haven't seen a good dial caliper with .001 resolution in my neighborhood for less than about $80, but I'm in the middle of a ha-tech industrial area where there is a ready market for such tools. It is likely that they are cheaper elsewhere. You can get cheaper ones, frequently plastic, that will resolve .01 easily. One that I picked up recently cost me $13 new.
The place to look in my area is at machinery salvage dealers and pawn shops. For a new caliper, try a hardware or lumber retailer.

Actually, a caliper good to .01 may be adequate for what is wanted here. Strings are made to the nearest .001 inch diameter, but usually you only need to know whether it's closer to, say, .012 or .018. If you've got a good eye, you can interpolate.

For a quick and dirty: if you can lay a short piece of the string down on a good flat surface, put the 1-inch mark of a 6 inch scale (ruler) on the string, push the "zero" end of the scale against the flat surface, and measure how high the other end is. The instant 6 to 1 multiplication will let you know fairly well what diameter you've got.

One of the more accurate methods is to put two pieces of the string between the jaws of an ordinary C-clamp, and use a feeler gage (like we used to use to adjust our valve clearance back when horses usually pulled us home). Pick a stack of feeler shims that fits the gap, and add them up. You can still get these feelers for $4 or so at most auto parts shops.
For accuracy, you look for a C-clamp with fairly large "faces" and clamp it loosely on a flat file. Pull the file through a few times to make the faces really flat, and then save that clamp for "precision stuff." You do need to use two wires, to keep the "swivel pad" square while you measure. (You can get "parallel jaw" clamps, but they're more expensive than we're interested in here?)
A wire gage can give you a fair approximation, and can usually be found for about $5 or less. The problem here is that only even "gage numbers" are given on most such gage wheels, and a lot of them cut off at #18(.0478) or #24(.0239), which is a little large for most strings. A few such gages can be found with thinner slots, but usually nothing smaller than #30(.012) or #32(.0097). A gage with #28(.0149) and #30(.0120) slots doesn't give much better resolution than what you can interpolate with the .01 $15 dial caliper.
For those who don't recognize it, the usual wire gage is a round metal "wheel," usually about 3-5 inches in diameter, and about 1/8 inch thick, with a bunch of slots around the edge. You find the slot that matches your wire, and read the size off the label on the wheel.
And for those who would quibble, the roughly 16 US (common) standard gages are different than the approximately 8 - that I know of - "Imperial" (common) standard gages, so your gage may have different numbers (inches) than mine.

The remaining problem is what to measure. "Light" and "Medium" mean different things to different string makers. The rule of thumb is that the "weight" of a string relates to the amount of tension needed to bring a standard length to pitch. Generally, measuring the core-wire diameter is appropriate, but the same core diameter may need significantly different tension if the "wrap" is different.

The end result is, your best bet is to put a known set of strings on, and if you like them, use the same kind next time. If you don't like them, then try something else. It's a learning process.

John