I'm well travelled, but Naples airport was the 1 place I'd wished I'd pre-booked a private car/taxi. I just felt the taxi drivers they were laughing and trying to rip us off!
That aside, I thoroughly enjoyed the trip and stayed in Naples itself. It was DH and myself- no children. I didn't wear any necklaces, just simple earrings and swapped my real wedding rings for a claires cheapy. Pickpockets can happen anywhere, but I'd heard of scooters stopping and pulling necklaces off.
From Naples, we caught a local train to Pompeii- it was straight forward and cheap. There are many 'tour guides' hanging around outside offering their service. We didn't use one, but you can say you'd like a 1, 2, 3hr etc tour and negotiate a price. You can of course get a booked tour from Naples if that is preferrable. Travelling there ourselves, meant that we could leave whenever we wanted. Ideally take a packed lunch. From memory, the nearest cafe was expensive and very busy. I got a map beforehand and roughly planned out the houses/things I really wanted to see.
A few stops on the train, heading back toward Naples is Heracleum. Smaller than Pompeii, but it could easily be done on a different day.
I've seen many documentaries about Pompeii and was expecting to see plaster cast bodies all over the place and frescoes. This isn't the case at all. Nearly every single mosaic and thing of interest is in the museum in Naples! Including many of the plaster casts. Yes, actually going to Pompeii itself was great- but I found the museum much more interesting. Note that there is a secret room in the museum where you need to be 18 to enter. It contains all manner of things with a sexual nature- a statue of man having sex with a goat, sexual frescoes and giant penises!
We stayed at Hotel Correra 241. Off the main road, down an alley way, at 1st it looked locked up- but it was safe, quiet and felt a world away from the hustle of Naples. It was down the hill from the museum, and 5mins to the train station- so a great location. Within Naples itself, we walked everywhere- museum, markets, restaurants etc.