Climbing to the top of the rock and back is a very rewarding experience, However to walk you should allow most of a day and be relatively fit to do it(My parents managed it it their 60's so not that fit).
Enter the nature reserve by entering just past the casino. Watch out on this section of road as it has no footpath and a lot of taxis use it.
At the pay booth either follow the road up the West side or follow the narrow path to the "Mediteranean Steps". You might have to ask where this path is as the sign is only small.
The "Medeteranean Steps" are the only path on the East side of the Rock above sea level. They give spectacular views over the sea and towards Africa but you need to be fit and agile to climb them.
If you take the road and head up the West side, stopping at St Michaels Cave. St Michaels Cabin (the cafeteria at the cave) does acceptable Tapas (a Spanish light snack) and is a suitable place for a rest before completing your climb to the top.
If it's open it is worth visiting O'Hara's battery, a gun emplacement at the uppermost Southern tip of the rock. This is also the point where the Medeteranean steps come out.
Head North along the top of the Rock to the "Top of the Rock Cable Station". En-route you will pass many fortifications from the Moorish to Napoleonic to the Present day.
From the cable station follow the road down returning past "Hayes Level" (an entrance to the Second World War tunnels and unfortunately closed at the moment) and the Moorish Castle to return to Gibraltar Town at the North End.