Where would you go for a day trip from rome with these specific wants?

Have you been to Assisi? I think it would meet all your wants.

I have not been to assissi, and a friend whos been to italy dozens of times has suggested it without my criteria. It does look like to cover almost everything (only one roman site but its the archeological museum which DOES look great)

Paestum

• 1) Should be a city or town - yup

• 2) Roman/greek/etruscan history sites - Greek -
  Originally Poseidonia. renamed by Romans, in Magna Graecia

• 3)Sites must be accessible by foot or transit - yup

• 4) I do kind of have a thing for churches - Greek temples any good?

• 5) great museums, specifically roman/archeological museums - Greek archeological museum, an excellent one, but not Roman, sorry

Underneath Piazza del Comune are Roman ruins. Underneath Santa Maria Maggiore you can walk down to the original Roman temple, below that is evidence of Etruscan ruins unnearthed after the 1997 earthquake. Some items from Roman times are quite visible without going down stairs or into a building. Churches? Dozens! The museums have very interesting items for the ancient history buff. YouTube has a few videos that I would suggest watching - they'll give you an idea for scheduling.
Now the bad news: it's about a 2 1/2 hour train ride each way and then a short bus/taxi ride up to the medieval town. I'd suggest, if you can't stay overnight, start early and take a late train back to Rome. Assisi in the evening is magical. Depending on when you are there, Assisi has a vibrant culture of music and pageantry.

Pencilstealer? I always 'borrow' the pens myself. :)

Nigel - thats a day trip from Naples, one that i Am doing while down there.

Consider going to Orvieto. It’s less than 90 minutes from Rome by train. The Cathedral is very impressive and the Underground Orvieto tour is quite interesting.

A city I never see on this forum is Grosseto. Expect no crowds in the museums (wonderful!), but check to see which days they’re open. We stayed there two nights several years ago. Here’s the quick copy of highlights. The walls are some of the best I’ve seen. Be ready to speak some Italian or have your phone handy to translate for you. We happened to be there when they had a night festival that was a fun evening of local talent of any type & age performing in various churches or piazzas, plus the museums were open late & free.

Transportation is a direct train that’s almost two hours. There’s some scenery along the water.

History and culture
Medici Walls:
The historic center is enclosed by intact 16th-century hexagonal walls, which you can walk along.
Etruscan and Roman past:
The city has a long history, with nearby ruins of the ancient Etruscan and Roman city of Roselle.
Museums:
Attractions include the Archaeological and Art Museum of Maremma and the Museum of Natural History.
Modern art:
The city features an open-air gallery called "Trame," where street art is displayed on shop shutters.

One of the museums:
https://maam.comune.grosseto.it/

It doesn't check all the boxes, but google Ostia Antica and see if it interest you. Easy to get there by local transport and foot, and should only take 3/4 of a day.

Orvieto

OP has already been to Ostia Antica

Pencilkleper, I wish I could say I've already been there myself but Tarquinia has looked to me like just a fascinating place to visit. It also looks like it checks all your boxes; take special note of the Etruscan Necropolis! Churches? Check. Archeological museum? Check. By train, you can get there from Rome in as little as 75 minutes, and it looks to have a local bus system for getting around town to spots that might be more of an effort on foot.

It has quite a nice website, including their local bus schedules, so take a look?
https://tarquiniaturismo.com/?lang=en
https://tarquiniaturismo.com/etruscan-necropolis/?lang=en
https://tarquiniaturismo.com/old-town/?lang=en#chiese
https://tarquiniaturismo.com/national-archaeological-museum-tarquinia/?lang=en
https://tarquiniaturismo.com/useful-information/?lang=en#arrivare]
https://www.eusepitrasporti.it/index.php?t=pag&id=106725

Oops. Missed that OP had been to Ostia Antica. Need to read before I leap.

There are a couple of museums in Orvieto whose collections include Etruscan artifacts, and the cathedral is magnificent. Orvieto is an excellent choice, but it has so much sightseeing you could fill about 2 days there if you could spare the time. Most rail departures take under 2 hours, but you need to add time to get to the top of the hill where the historic center is located. There's a funicular, which I think may run every 15 minutes, or alternatively a bus (which is less fun). Definitely a day trip for which you'd want to get an early start.

Assisi is also lovely, as long as you start at the untouristy top of the hill and slowly wind your way down to the magnificent basilica. The immediate surroundings of the basilica are full of shops selling tourist schlock (albeit of the religious sort). But Assisi is even farther from Rome than Orvieto is. The morning trains all take longer than 2-1/2 hours each way.

I trust you've been to the Etruscan Museum at Villa Giulia in Rome.

Robert - ya I took a better look and it looks great!

Philip - ive had my eye on orvieto. It does tick all the boxes. Its a nice city too.

Jean - now this is one i havnt heard of, and at first glance it looks pretty cool! I like off the beaten path places. I'll need to take a better look.

Kate - i ALWAYS forget about Tarquinia. Then i watch some doc on rome and it always comes up. It definitely ticks all the boxes and might actually make up for not seeing Tivoli.

Craven- actually I have not! I only had 7 days on the ground my last trip to Italy (first euro trip) . Im staying relatively close so that might actually work out.

Have a look at Viterbo. Checks all the boxes. Off the main tourist radar.

Brad