For our application, Type K is the best suited, and most economical. (Type R and S have a combination of platinum/platinum rhodium that is very, very expensive and have a range of 0 to 3100 degF.)
Most multimeters cannot read T/C inputs. Only certain digital multimeters have the option of a thermocouple input. Thermouples have an output of microvolts (millionths of a volt).
A relatively inexpensive controller can be had at the websites mentioned in the above posts and that is probably your best bet.
Craig