Which process for proposing an amendment is easiest and which is the most difficult? which process for ratifying an amendment is easiest and which is the most difficult?

The easiest should be getting a proposal by 3/4 of those in both houses of the congress. This should be the easiest because you don't need 3/4 of all members elected, but rather the 3/4 of those who are there if a quorum exists. This means that a high majority of the quorum is needed and it is not difficult to get all these people to be there and to vote for what your party wants if you're in the majority. 

The most difficult should be the second option which requires the state legislatures of 3/4 of states to accept the proposal and send it to congress. This means that almost 40 states have to adopt the proposal in their own state legislatures which is extremely difficult to do and is extremely time consuming. 

When an amendment proposal is adopted, then the easiest way is for the 3/4 of state legislatures to ratify the amendment before it becomes a part of the constitution. This is a very time consuming process but it has been the way for the amendments to become ratified almost every time in history that an amendment proposal was adopted

The second, more difficult option, should be forming state conventions in which the conventions have to ratify the amendment proposal instead of the state legislatures. This was only used once in the history of the United States. Which mode of ratification would be used depends on what the congress chooses as allowed by the Supreme Court.