Norwich day trips less than 3 hours from city center- No churches or museums

Do you like heritage railways? You might consider the Poppy Line steam railway out of Sheringham along the coast? Cromer on the coast, famous for its crab, is also nearby. Since you enjoy being outdoors, the Norfolk Broads are very near to Norwich.

using a car or the train?

We are staying in Norwich to see my daughter who's doing a semester at UEA.
2 hour train is ideal for a day trip as we have a flat in Norwich that is paid for for the 4 nights.
We are constantly finding info convincing us to cancel our car reservation since we have never driven right side and that New Yorkers in particular find it particularly dangerous.( sorry fellow New Yorkers) Stress free is preferred! I have no idea if the Ubers or trains or even a local car for hire driver is a feasible option. We haven't done much planning due to home obligations. I'm so glad for the suggestions on this thread!

Driving in East Anglia is no fun. I know - I have to do it frequently. There are no motorways in East Anglia (the M11 finishes at Cambridge, the A1(M) is barely a motorway and has very short segments, and the A14 is no motorway). Most of the roads are single carriageway, with rare bits of dual. The roads are clogged with commercial traffic and that is required by law to go significantly slower than cars. The only way to make any sort of time in East Anglia is by passing those pesky trucks. Not much fun if you aren't used to driving on the left.

I don't know when your trip is, but if in summer you may enjoy the broads, or heading up to the coast and perhaps having some fresh Cromer crab.

While this is not truffle hunting, you may be interested in visiting Blickling Estate. It is about 30 minutes drive from Norwich. We did this trip last summer and hired a local taxi that dropped us off and then came back to pick us up at an agreed upon time.

The estate was beautiful and there was a nice gift shop and small cafe on the grounds. The gardens were also quite nice for a walk around. The home was owned by the Boleyn Family and is rich with history. It is a lovely place to spend a day or several hours.

When are you visiting? The fungi foraging season is best April/May and September/October. It is quite a trendy thing to do, so most courses get booked up months in advance. Due to the weather, it hasn't been a good autumn for fungi, apparently.

There are plenty of places on the coast for seafood, including Cromer and Cley (pronounced to rhyme with rye).

Look, you're not wrong about driving on the right. But you're shooting yourself in the foot by declining a car. This area is filled with half-timbered villages, historic estates and mansions, nautical-related destinations, historic pubs, and beautiful landscapes. Most of those destinations are not next to the train station, even if there is acceptable service to a town.

Only you can decide what is safe for you and your family.

I would think that truffle hunting or mushroom foraging would involve a guide or concierge. You could easily arrange for the guide to include transportation - you just have to allocate additional money for the outing. Is that an acceptable trade-off for you?

Most Americans (you left your home area off your Profile, so we have no idea .... and you didn't give the time of year for the trip.) have excessive confidence about their supreme ability to drive. (Irony intended .... ) I also don't like driving on the right, and my wife is assigned to keep saying "Keep right, keep right." Our host here, Rick, has written about the subject, as have many other travel writers. You have to decide what's best for you.

We enjoyed the Norfolk Broads, and some arranged, captained and crewed, sailboat excursions (even though we never boat at home) because they were related to the (fiction) children's books written by Arthur Ransome. The Museum of East Anglian LIfe is charming, and has quite a bit of walking. Remember that it rains a lot in England. We enjoyed Cambridge very much, worth a full day.

We are from the suburbs of New York and we are going soon in October 2016. Thank you for the info/insight