First Winter Trip: Good Side Trips from Rome/Naples/Salerno?

Gosh that is tough, I look forward to following along to see what I learn.
I don't know if you dig archaeology as much as I do, but in Ascea, an hour south of Salerno, there are some minor ruins in an otherwise unremarkable town. We enjoyed a rainy day there several years ago. The train skirts the Cilento coast except for Agropoli, which might be worth checking out on a sunny day.

Hi Ann! What about (the palace of) Caserta as a side-trip from Naples? I would be sure to give Naples enough time (most people who hate it did not give it more than 2 days, if not less). Also, I would seriously consider Capri in the off-season, it may be much more satisfying than when it's overrun.

Do you already have Paestum and/or Herculaneum in your plans?

I'm aware of Pompeii, Ercolano, Paestum and Oplontis. Probably won't go to all of them because, in truth, ruins pretty much bore me except for the mosaics (some of which are now in the archaeological museum, I know). I've done Ostia Antica and didn't find it any more interesting than I thought I would. (I used Rick's audio guide, so I wasn't just wandering around aimlessly--not that I'm blaming him.) I'll do a couple more such sites (almost certainly including Pompeii) more out of a sense of duty than anything else. I figure it would be nuts to have as much time in the area as I will, along with the cool weather, and not take advantage of it. I may try the Mondo Guides tour of Pompeii if it's being offered while I'm in the area.

Caserta might be an option; I've heard of it but don't know anything about it; I'll put it on my list and do some research. My problem is that when it comes to decorated interiors, my preference runs to late-19th-century and 20th-century stuff, so my reaction to most palaces is not that different from my reaction to classical ruins. At least in the palaces you aren't exposed to the weather, though.

I sometimes stop in at a big-city tourist office and ask about a side trip with nice scenery; I may need to do that here.

The Paestum museum was quite impressive, so be sure to look into that even if you are meh on temples.
If you are this jazzed about Pompeii, you can probably scratch Velia (the site in Ascea) off the list (though the museum did have some nice pieces).
Vietri seemed large enough that at least some of the shops may be open. And it's so easy from Salerno that I'd try anyway. Procida and Ischia, too.

Thanks for all these ideas. I'll update my sightseeing notes. At the moment the weather forecast is in line with historical averages. That will be great if it continues; I like to do an indoor sight and then walk around outside every day.

I found the 3 Greek temples at Paestum highly impressive, said to be the best specimens of Greek ruins anywhere, including Greece. From 550, 500 and 450 BCE, as I remember. Especially beautiful in the late afternoon light. I agree that the small museum by the ruins is excellent, with mosaics, frescoes and other artifacts. A highlight for me was a section which showed how the temples looked when in active use, painted with bright colors, instead of just gray stone like today. Very different and very memorable. When I was there, several years ago and pre-pandemic, the temples were accessible every day, but the museum was closed one day a week, Monday I think.

That's a good point about digging into the details about each site's hours of operation. It's easy to miss things like that at a glance, and a mid-winter trip probably means abbreviated hours as well as more closing days.

As far as Ischia and Procida go, would either of them be a decent day trip destination? I've read enough about Capri to know I need to spend multiple nights there--probably just three because of the cost--when I return some spring. Would one or both of the other islands be worth doing in the winter (limited daylight) from Naples? (I know I need to check the ferry schedules to be sure service runs early enough and late enough to provide a decent amount of time on the islands.)

OK, really nice town for you: Sant'Agata de' Goti. We stayed there for 3 nights at one of those rare in-town agriturismi (because they run a vineyard), Agriturismo Mustilli, in an old palazzo with breakfast, and my husband took a cooking lesson there, and we visited the family's wine cellar (also in town). The town was fun to wander around in. It's an hour and 40 minutes by bus from Naples.

Sorry, I forgot how to make a short link, but here's some photos of it perched on its steep cliff:

https://www.google.com/search?q=sant%27agata+de%27+goti&tbm=isch&ved=2ahUKEwjC7L3Vx_r8AhXzPd4AHe_hD_4Q2-cCegQIABAA&oq=sant%27agata+de%27+goti&gs_lcp=CgNpbWcQDDIFCAAQgAQyBQgAEIAEMgQIABAeMgQIABAeMgYIABAFEB4yBggAEAUQHjIGCAAQBRAeMgcIABCABBAYOgQIIxAnOgYIABAIEB46BggAEAcQHlClB1iiDWCvIGgAcAB4AIAB5wGIAdYDkgEFMy4wLjGYAQCgAQGqAQtnd3Mtd2l6LWltZ8ABAQ&sclient=img&ei=AKDdY8LsHfP7-LYP78O_8A8&bih=579&biw=1273&rlz=1C5CHFA_enUS507US521

There is no duty to go look at ruins. If there was duty, I would have ridden in a gondola, gone to the top of the Eiffel Tower, seen the Last Supper, visited Buckingham Palace when I lived in London for 4 years, and also quite a few things in Washington DC (my birthplace and childhood home).

We are big Naples-in-winter fans. My theory is that whatever "grit" and "chaos" people complain about there is just not a problem in winter. We were 5 nights there and should have made it 7 in order to see what we wanted to see.

Easy to get to Tarquinia by train from Rome, maybe an hour and a half. Or Prontobus straight from the airport in an hour (it skips Termini). We enjoyed 3 nights there last September ( even though it was TOO HOT then!)

Why Salerno?