I am having trouble understanding the difference between a 'necessary' and a 'sufficient' condition (in philosophical usage). Would I be right in thinking that the former is a condition that must be present in order for something to happen, while the latter is merely 'enough', i.e. that no other condition needs to be met (while with a necessary condition others can be met)?

An example might help you understand the difference. Being at least 35 years old is a necessary condition for being President of the United States. You cannot be President unless you are at least 35 years old. But being at least 35 years old is not a sufficient condition for being President, i.e., to be President, it is not enough that you be at least 35 years old. Otherwise, I would be President.

How do you tell the difference between a reductio and a surprising conclusion?

The conclusion of a reduction need not be surprising. Suppose I am trying to prove that 2+3 =5. I might proceed as follows: Assume that 2+3 does not equal 5. Since 2 = 1+1, and 3 = 1+1+1, that would mean that (1+1) + (1+1+1) does not equal 5. But (1+1) + (1+1+1) does equal 5. So our original assumption must be false, in other words, 2+3=5. This example isn't that interesting, but the point is to show that you can use the reductio strategy without ending up with a surprising conclusion.