Can someone provide a trivial example to the "reduction" procedure used to prove hardness?
First of all It may be a good idea to read about SAT and Clique problems. Quite a good point to start your research in NP problems. I'd suggest investigating SAT-3CNF, as it is a flagship example of an NP-Complete (both NP-Hard and in NP itself) problem.

Now, to show that an NP-Hard problem X can be reduced to Y would indeed prove that Y is NP-Hard too. To do that you need to provide a relation that maps X to Y (so for any example from X this function maps it to an example from Y.) What you basically want to show is that being able to solve Y efficiently, you'd be also able to solve X efficiently. Personally I believe you should start by reading about SAT-3CNF and Independent Set problems - there are some great examples explaining how they are equivalent, and the mapping is pretty intuitive. You may read about them, for example, here.

To answer your last question - I believe dealing with NP problems can be treated as non-trivial itself. Therefore I would recommend to spend just some time with them, get used to them and try to understand them, so that you are not just pointlessly trying to memorize problems and solutions.