I recommend walking to the cemetery down the steep slope of Swains Lane beside it, so that you can see the area that the legendary ‘Highgate Vampire‘ hangs out in (plus you do not want to walk up this hill for anything- even the flavour of the month vampire). When you leave you can go back through Waterlow park it’s an easier walk, plus they say it’s a less refined version of what the cemetery looked like when it was first created.
You have a choice of two parts for your visit- it is better to plan around the tour times for the West side as they are fixed and you must go with a guide. That side is £7, the East side you can wander about on your own, and it is £3.
The East side is the celebrity hangout, not like you will see celebrities meandering about, but you can see where they are buried! Karl Marx is the draw card, but Douglas Adams and Michael Faraday along with a fair chunk of Charles Dicken’s family (not the man himself though- he’s in Westminster Abbey).
The West is a great trip through the history of London, and depending on your guide, is an insight into the personal histories of Londoners who we have forgotten over time, or their architectural tastes and preferences. You can ask to visit certain areas if they have time, but they do cover the Lebanon circle and Egyptian Passage, as well as the first grave. I would recommend asking about the history of cremation and to see Thomas Sayers, the bare knuckle fighter (he does seem to be a part of the regular tour though).