π: Ctrl+P or
Alt+P or pi.
∞: Ctrl+8
or Alt+8 or infinity.
θ: Ctrl+T or
Alt+T or theta. You can also use x or t instead of θ in polar mode; they will be
replaced by θ when you press Draw Graph or Enter on the keyboard.
The quickest way to type domain= (-infinity, infinity) is by deleting the domain entirely (including domain=) and pressing Enter.
To graph a piecewise-defined function (or parametric curve) use the multi-graph window. Type in the number of pieces, and enter each piece with the corresponding sub-interval as a single function.
When drawing, particularly, polar or parametric graphs on an interval you can use the multi-graph window to see portions of the graph with different colors. Simply subdivide the interval and enter the same formula with each sub-interval as a single function.