Different nuances of words such as 进行, 实施, 执行, etc
Good question. In fact, you can hardly precisely distinguish them by only looking at their definitions. Some are even used to define each other in the dictionary. The best way is to remember some very common collocations containing them. This is actually how native Chinese speakers use these words. Usually they are used only in particular collocations.


进行

This is used to describe that something is currently happening, or is used as a transitive verb followed by formal activities.


  会议正在进行中 = The meeting is going on right now.
  
  进行科学实验 = conduct scientific experiment

实行，实施

This usually means that you already have a plan or a theory for something and now you need to implement it or carry it out. I think they are interchangeable in most cases.


  实施办法、方案、计划、政策
  
  实行道路管制 = traffic control (I can't find a verb in English)

行使

This is pretty much only used for legal rights.


  行使权力，行使职权

落实

This is similar to 实施. Literally it means you want your plan to REALLY happen. 落实 compared to 实施 is like "fulfill" compared to “implement, carry out". There is a phrase 落到实处，where 落到 literally means "to fall into (somewhere)" and 实处 means "a real, tangible place" (as opposed to "in vain"). Some examples:


  落实政策


落实 can also mean "determine" or "make sure", but that's another story.
执行

This refers to the actual action of 实施 something. Loosely speaking 执行 refers to a more concrete, specific action.


  我们打算明年实施这个政策，在具体的执行过程中遇到的问题要及时向我汇报 = We plan to carry out this policy the next year. During the execution phase if any problem comes up you need to keep me posted.
  
  执行任务、命令



You may want to know that:


This is not a comprehensive list.
By providing the list I am not saying that you have to stick to it or stick to existing collocations. But if there is already a collocation that fits what you have in mind, using it can minimize your chance of sounding unnatural.
To find more such collocations you can download a Chinese input method and turn on the collocation option.