"Nevertheless", "on the contrary", and "on the other hand" have different meanings and uses. "Nevertheless" means "in spite of that" or "even though the previous statement is true." "On the contrary" means "as opposed to the previous statement." "On the other hand" means, "from the other point of view." "However" is a more common conjunctive adjective than "nevertheless" and "however" has more meanings, so I have changed "nevertheless" to "however" in your sentences. In your sentences, "however" fits best because it has a general meaning of "by contrast." "On the contrary" means that the two clauses are opposed, and not just different. "On the other hand" is used when there are two different points of view that can be considered. These conjunctions can be hard to understand, but they are important when making your writing follow a logical order. 1. I used to have a lot of friends at school; however, I have just a precious few at work. 2. Most TV programs in the past were a bit dull, with very simple plots; however, today's programs are more interesting. 3. If they have some free time, many women would choose to hang out with friends to go shopping; however, most men prefer playing sports. 4. Prior to my marriage, I was almost always on my own when I traveled; these days, however, I always consider taking my family with me during my trips.

These are very good efforts and you show your determination to understand the usage of "nevertheless". However, you need to use "however" instead of "nevertheless" in your examples. However is used to introduce simple contrasts in information, in your examples between the past and present, and between men and women. "Nevertheless" or "nonetheless" is used to introduce a positive aspect of information which is generally negative e.g "Most TV programs in the past were dull, with very simple plots; nevertheless, they were very popular with viewers." There are a few other little mistakes in your sentences but nevertheless, they are very good!