Train from Naples to Pompei and then to Salerno

I think it's six one way and half dozen the other, but to help you decide: The walk from Pompeii site to the Trenitalia station is about 15 minutes (you can map it on Google Maps--the train station is Pompei {one i}, to the Anfiteatro entrance). I came without luggage, but for the average person, the walk to the Trenitalia station is basically nothing. It is only slightly taxing if there is heat or abundant luggage involved.
How many of "you" are there? Kids, etc.? Will it be in summer? All of those factors are important. You could consider using a taxi.
In short, I would cement the luggage storage options and proceed accordingly.

Or you could taxi one or both ways

thanks both. There will be 2 of us and no kids and we're pretty fit and have no problem walking, though we will have a mid-sized case each and a smallish backpack. Just to make sure, when you say it is 15 minutes walk from Trenitalia to the site do you mean to the Anfiteatro (Ampitheater )entrance? Does anyone know how far it is to walk from Pompei Scavi train station to Pompei Station? If too far approximately how much would a taxi would cost? Also, just to make sure, is the station on the Trenitalia train line actually called Pompei Station? I want to make sure if we choose this way to get there we get off at the correct station!

Just to make sure, when you say it is 15 minutes walk from Trenitalia
to the site do you mean to the Anfiteatro (Ampitheater) entrance? Does
anyone know how far it is to walk from Pompei Scavi train station to
Pompei Station? If too far approximately how much would a taxi would
cost? Also, just to make sure, is the station on the Trenitalia train
line actually called Pompei Station?

Google Maps confirmed my memory of a 10 minute walk (I said 15 to round up) from Pompei, the Trenitalia Station, to the Amphitheater entrance.
The Circumvesuviana line stop is Pompei Scavi - "Villa dei Misteri". There is also the Campania Express, same stop (fewer other stops).
I recommend zooming in on Google Maps to get a sense of all of this. It looks like about 20-25 minutes to walk between the two stations.
Rometo Rio might give you a taxi estimate, but I wouldn't sweat it--practice your Italian just in case so you can clearly say where you want to go. There is probably a bus as well, but that could be a pain with a suitcase.
{To confuse matters, there used to be two Circumvesuviana stops at Pompeii, but it appears from the Circumvesuviana website that there is now only one, at Villa dei Misteri.}
Hopefully someone will chime in with clarity on the luggage storage, as I have seen conflicting info and so can't comment on which is preferable.

You don't have to guess any of this. Google Maps is your friend. Month and year of trip?

I would expect it to be sweltering hot with high sun from June to September. I would definitely look for a modern-Pompeii tourist web page to get some taxi phone numbers, but maybe the big ticket office at Porta Marina will call for you if you ask nicely? You will be absolutely exhausted even with only two hours inside Pompeii, on the ancient stone streets radiating heat back onto you. You will not feel like walking to the Trenitalia station, even from Amfiteatro. (I have not done this.)

Someone else's post mentioned a bus station (coming from the Naples airport) near the Porta Marina entrance, shows on Google Maps. I wonder if a low-budget traveler might find something useful at that bus station, I mean like a modern-Pompeii city bus?

I have not stored luggage at Pompeii (we daytripped from our Sorrento hotel room), but no one posting here has ever reported a problem with the ancient site's attended luggage counters. I would assume Amfiteatro is just as good if you end up checking bags there. I think it might even be free. Luggage is not permitted inside the site.

https://community.ricksteves.com/travel-forum/italy/pompeii-train-station

I would normally not worry at all about walking between the two Pompeii train stations as long as my heavy luggage was wheeled, but in addition to the heat at Pompeii Scavi (if this is a summer trip), you will have been walking on streets paved with some of the world's largest cobblestones. It is exhausting.

Taking the train from Naples to Pompeii and then to Salerno sounds like a great plan! I recommend checking out dbfahrplan.com to get the most up-to-date train schedules and connections. The journey from Naples to Pompeii is usually quick, about 35 minutes, so you'll have plenty of time to explore the ancient city. From Pompeii to Salerno, the train ride is around 30 minutes, giving you a chance to relax and enjoy the scenic views along the way. Make sure to check the timetable in advance to avoid any surprises.

Thanks all, great information. It makes sense I think to take the TrainItalia Napoli Piazza Garibaldi train to Pompei Station given that's the station we'd need to be at to travel on to Salerno