You could answer your questions with a thermocouple, I agree. You can order something from omega.com where they have many options. They are the Sears of heat and temperature, I feel.
First, I suggest Type K. The “Type” is a reference to which two alloys are chosen for the two legs or wires of the thermopile. It’s not the junction, but along the length of these wires, that the voltage signal is generated. All the types have their merits, but K is a good all-around choice, and it is highly available, so more options are available.
You described wanting a handheld display meter and a thermocouple with a rigid sheath and flexible cord. That kind of sheath is typically a “magnesia insulated metal sheath” or “MIMS” construction. They’re generally available with the junction exposed, or hidden inside the sheath and not connected to it, or hidden inside the sheath and connected to it. In experiments I’ve found the hidden and not connected type to be the least noisy, but they’re all OK. Hidden and not connected is generally called “ungrounded”, and connected called “grounded”, though this is a misnomer unless you connect the sheath to ground. Supposedly the exposed junction can respond faster, but they’re still sufficiently strongly thermally coupled to the sheath that they don’t finish responding faster, they just start responding faster. They’re also kinda vulnerable.
May I also suggest you get some insulated wire thermocouples that are not sheathed? They respond much faster, they snake into little places, you can tape them up better than a sheathed one, and they are quite inexpensive. At Omega you can get a convenient five pack for maybe $40 or so – what a great stocking stuffer!
I use thermocouples quite a lot. I buy types K, E, and T mostly. I buy 20 gage to 40 gage insulated wire ones, with “Teflon” insulation and with fiberglass insulation (for different temperature ranges). I also buy sheathed ones, often 1/8" or 1/4" or really small ones like 10 mil. I have simple readouts from Omega and Fluke and Hart Scientific. I also have data loggers and PC-connected data acquisition boxes that take them, including from DataTaker, National Instruments, Squirrel, Hobo, Keithley, and a few others. And I do a lot with platinum and nickel resistance temperature detectors and resistance thermometers, and sometimes thermistors. Thermocouples are weird, but thermocouples are nifty.
It’s true, you could also get an infrared pyrometer, or for that matter a thermographic camera. I have several of these, including cameras from FLIR. They certainly have their place – but bear in mind that with infrared methods you have to know the emissivity of the thing you’re measuring, and that’s often harder to find than the temperature was in the first place.