As with traveling many different routes in Los Angeles throughout the day, how close is one's destination really isn't a matter of distance, but a matter of time! With a city like Cleveland, or Tampa Bay, you could answer this question like "downtown is 3 miles away", or "the stadium is 3 exits down the freeway".
However in Los Angeles, travel times vary so vastly, given the time of day one is making their journey. While Beverly Hills is not far from Universal Studios at all, it could take an hour to get from one to the other depending on the time of day you are traveling.
In terms of action... I love the daily bustle that goes on over at Rodeo Dr, Canon, Roxbury, etc. At night this area turns into a ghost town. You could travel slightly east on Wilshire and will find lots to get into. One of my favorite restaurants is in this area... Kate Mantilini! (Kate Mantilini happens to be from my hometown, Cleveland) This is where they shot a well known scene for the movie Heat. When I discovered this, I was sitting at the exact table that Al Pacino and Robert Deniro were having that intense conversation in the movie!
All the best,