Has anyone taken the cable car to the top of the Rock of Gibraltar and walked down and see the sights on their own or do you suggest a private tour?
Yes - I have done the cable car and walked down. Also, I have walked up and then walked down. 2019. Things can change, but I can give you my impressions.
1) The cable car can get very busy if there are cruise ships in port. I was lucky not to experience that. I am sure I didn't have an advance ticket, but I don't know if that might not be prudent currently if there when cruise ships are in port.
2) There are multiple "sights" on the Rock. You might want to identify if you wish to see St. Michael's Cave, the Skywalk, the Apes, etc. You can easily google a route based on your interests. It isn't hard to explore the Rock on your own.
If you choose to see the Barberry Apes (actually they are macaques), do be watchful. I had had multiple experiences with wild macaques in several Asian countries where I lived and frequently hiked. Despite my care as I walked through their area, one of those Gibraltar critters grabbed at my day pack and was almost successful in getting inside. He would have been disappointed - no food.
3) The Mediterranean Steps is a steep hiking trail. I don't like heights, but had no trouble going up. I, personally, would not take it going down. Other than this route, everything else I encountered was paved and unchallenging even with ascending heights.
4) There are some very interesting WWII sites. Some are batteries that can be accessed without admission, the WWII cave installations require a ticket. There was an audio tour when I visited. I pre-booked this.
I don't know if you are getting off a cruise ship or on a guided tour to the general area, or if this is a DIY trip. If you are willing to do a bit of research and have no mobility issues, exploring by yourselves is quite easy. If either of these apply, I would look into a tour that would drive you to starting points for the WWII caves or other areas of interest.
Enjoy - I found Gibraltar quite interesting.
We just did that in September. No need for a tour. In fact, I would avoid the taxi tours like the plague.
Walking down was great. Our route was Top Station > Skywalk > St Michael's Cave > Queens Balcony > Apes Den > Old Queen's Road > Union Jack Steps > Town. We didn't do the Windsor Suspension Bridge but it close to that route, so you could add that. Views were great. Explanatory signs were at all the sights. Apes were at the Apes Den, but there were many more at Top Station.
Advice: make sure you get your ticket for the cable car in advance. We had a reservation and still waited about an hour to get up. Without a reservation you would have TWO lines that long. Get the One Way Fare (https://naturereserve.gi/tickets/).
Word of warning: The walk is moderately steep. My wife ended up with a sore knee. Take it slow, enjoy the views.