How can I see what I just did on my computer? For example, an activity log showing that I copied a file from here to there?
Not at that level of detail by default. Logging is available for many actions by checking out your
console app and taking a look at contents, but it's down to each app to decide what level of system logging is needs to do to be useful. Increased logging is very useful to aid in debugging and to understand user problems, but at the cost of the management and performance hits of dealing with enormous log files. In your case, the finder doesn't log each individual file creation or amendment, but you may be able to find other finder replacements which have customization debug level logging to achieve this.