Basics: A thermocouple is just 2 different metals joined together. When the junction is heated, it produces a voltage which depends on the temperature and the metal compositions used. Any metal combination can be a thermocouple.
A number of particularly useful compositions have been developed and their temperature/voltage characteristics compiled into standard tables. They have then been given letter type codes. Different types have different useful ranges.
There are several thermocouple types that could be useful to "us". The most common, and probably most useful, is type K.
The metals that form the junction in a type K are Chromel and Alumel. The tables go up to 1372 degC/2500 degF and down to -260 degC (-436 degF).
In most cases, the other materials used in the thermocouple assembly will have a maximum working temperature below the maximum working temperature of the junction.
Thermocouple wires need to be insulated from each other so that they only touch at the junction. If they touch again elsewhere, the point of contact becomes a new junction and this will louse up the reading.
Ceramic insulated assemblies are good for fixed thermocouples. They usually have sections of double-barrelled ceramic tube threaded over the wires and are not rigid enough for horizontal insertion much beyond a couple of inches or so. Putting them in a tubular ceramic sheath provides rigidity, but at the expense of response time; lots of thermal mass makes them slow to respond to changes in temperature.
Mineral Insulated thermocouples consist of the 2 junction wires embedded in compacted Magnesium Oxide powder within a metallic sheath.
By some form of near-magic, the MI cable is unrolled, the junction welded and the end of the sheath welded shut by the company that makes your thermocouple to order.
The sheath material is usually the limiting factor for the maximum working temperature of MI thermocouples. In many cases, the sheath will oxidize at high temperature and the oxide will flake off at low temperature, so the sheath gets thinner each cycle. It is pretty thin to begin with. For our purposes, type 310 stainless steel and Inconel are both considered good to 1100 degC (2012 degF). Omega have a proprietary material, Omegaclad XL, that is claimed to be good to 1335 degC (2435 degF).
MI thermocouples are reasonably rigid and can be had in a number of diameters. The most common type of junction is an insulated junction, with the junction completely surrounded by the MgO powder. This has fairly slow response. Exposed junctions just weld the measuring junction and cut back the sheath leaving the junction exposed. Response is fast, but they are fragile. Grounded junctions weld the thermocouple junction to the closed end of the sheath. This gives a reasonably fast-responding, but reasonably rugged, assembly. Not all combinations of sheath material and thermocouple type are suitable for grounded junctions.
My opinion/suggestion.
Omega KHXL-14G-RSC-24 Themocouple assembly.
It is a 1/4" diameter Omegclad XL type K with grounded junction, 24" long below a handle, fitted with a curly cable and a miniature plug.
24" long may seem excessive, but it will not seem it when you are trying to see how the temperature changes throughout your forge. 1/4" diameter is rigid enough to reach the back of most forges without sagging too much and the speed of the grounded junction is also helpful when you are trying to see how the temperature changes throughout your forge.
http://www.omega.com/pptst/KHXL_NHXL.html
Enter the part number in the search box for a price; the part no builder only gives a maximum of 18" and the extra 6" is worth it IME.
Then search ebay for a TM902C handheld readout and order one (preferably two). Price should be around 5 bucks delivered. If you only think in Farenheit, you'll need a calculator or conversion table. Alternatively you can pay several times the price for something that reads in degF.
If you need hotter, things get expensive