Hope and wish are synonymous but have different root meanings and inferences, therefore they cannot be used interchangeably all the time.
I think the best way to distinguish them is that "a hope" is a feeling, whereas "a wish" is an expression of your hopes. In idiomatic use, the difference between the two has blurred a little, but traditionally, you make a wish while you have a hope.
Here is an example where they are interchangeable:
I hope for good weather
I wish for good weather
In your example though, they are not interchangeable:
I wish you best luck in pursuing the Ph.D.
This is fine and idiomatic.
I hope you best luck in pursuing the Ph.D.
This is not idiomatically correct. It should be:
I hope you have good luck in pursuing the Ph.D.
Your last example has a different idiomatic meaning:
I wish everything would go smoothly for your research.
This would be taken as a lament - when we use "wish" about something that is already happening it suggests that we are wishing for a reversal of fortune or a change in present conditions. I would assume from this statement that things are not going smoothly at present.
A more idiomatic way to express a desire that things continue to go well would be:
I hope your research continues to go smoothly.
I wish you well in your continued research.