I've made that trip a couple of times, also once in a PA28.
For sightseeing, I follow the south coast from Bournemouth on the way there and the North Coast to Bristol on the way back.
Nothing special to note.. The danger/restricted areas are not an issue on the weekends. The Bristol CTA forces you out over the water of you go around it but the view of the docks is quite nice.
Coming from Elstree, I did not refuel the PA28, but coming from further away you may need to, so best to fuel on arrival, giving you more flexibility when leaving.
Proximity of the cliffs and the likely winds means some potential fun on final, but the runways are plenty long to allow for that.
Lovely walk over the cliffs to St. Agnes for lunch if you want it.
Should be a lovely trip