What are the steps for solving this equation on a handheld calculator? Where should I start? I know many calculators have functions to do this quickly, but this is more a question about using a calculator.
$$1199.10 = 200000\left(1+\frac{0.06}{12}\right)^{360} \frac{0.06 / 12}{\left(1+\frac{0.06}{12}\right)^{360}-1}$$
Answering this
With a complicated expression like yours I would compute some of the intermediate ingredients separately and write them down (or save them in the calculator's memory if it has one). Then combine the ingredients.
So $$ A = \frac{0.06}{12}, $$ $$ B = \left(1+A\right)^{360}, $$ $$ C = 1/(B-1), $$ $$ 1199.10 = 200000 \times B \times A \times C $$
Along the way you can think about the intermediate values and make sure they seem reasonable. If you do the calculation all at once it will be harder to detect simple typing errors.