Does one need to know Morse theory to learn Discrete Morse theory? How much of Milnor's Morse theory is essential? Does one also need a background in differential topology for discrete Morse theory?
Does one "need" to know Morse theory to learn discrete Morse theory? No, go for it. But if you lose motivation along the way, wondering what it's all about, here's some further thoughts.
Does it "help" to know Morse theory to learn discrete Morse theory? Yes. The mathematics needed to understand both the proofs and the applications of the original Morse theory are simpler than for the discrete Morse theory, and it is very good practice to read the simpler theory first. Basically, the original Morse theory is just based on some simple principles of multivariable calculus, and one gets to the applications very quickly.
One other point: Milnor's Morse Theory is one of the best books in mathematics. Period. If you have the opportunity to read it, you should eagerly jump at that opportunity.