day trip to a village from London?

Lewes. Bromley.

Ely in Cambridgeshire is about an hour away by train. Bury St Edmunds is not too far from Ely. Both are beautiful small towns with cathedrals.

Trains will mostly take you to small towns rather than villages, and you will then have to take a bus. Or consider the Bluebell Line steam trains, now easily accessed from East Grinstead station http://www.bluebell-railway.com/directions/by-train/
I used to live in West Hoathly which is exactly the sort of village you're looking for, and still tucked away off any major road with plenty of local footpaths, an old church and 15th century priest's house (with museum), and the Cat Inn, which does a good lunch (not sure about a tea room). Train to Three Bridges and then 20 minutes on no. 84 metrobus (metrobus.co.uk). N.B. This bus doesn't run on Sundays or public holidays.

Suggestions in the first two posts are towns, not villages. Lewes sounds like the kind of thing you are looking for but bigger, but Bromley is a larger town that IMO lacks the character that you are seeking.

The trouble with the characterful village that you seem to be seeking is that these places tend to be quite a way from rail stations and public transport in general.

I would suggest Headcorn in Kent - a direct rail link from London Charing Cross, London Bridge or Cannon Street. The village has a couple of tea rooms and pubs and plenty of independent shops. Sissinghurst, one of the best gardens in England is a short bus or taxi ride away. In the opposite direction from the village, north of the rail station is Headcorn Aerodrome, where a Spitfire is based (you maybe lucky and see it fly) and there is a good free air warfare museum there. You can take a 15 minute flight in a Tiger Moth, booked well in advance for £135 (flying the Spit is an eye-watering £2,500!). Wings Bar there is popular.

Alternatively, Robertsbridge, Wadhurst or Wye would suit, as all are characterful villages with a range of shops and places to eat and walk on a direct train line from London.

Headcorn is good, or Chilham near Canterbury, which is on the line from Charing Cross.

Here is an idea. We just took a day trip to Eastbourne in East Sussex from London and walked up the South Downs Way from Eastbourne station to Beachyhead. Our original goal was to walk on to Birling Gap and then to East Dean but I was coming down with a cold so we had to truncate our trip and stopped at the Beachy Head Inn for lunch and a cup of tea and then got a taxi back to Eastbourne station.

You would take an easy train ride to a seaside town and then take a medium intensity walk along seaside chalk cliffs past two light houses and then on to the town of East Dean. You can get a bus at several places near the walk if you need to cut it short. We didn't get there but the town of East Dean looked to be pretty quaint from Google Earth and pictures I saw during my research. There even is a sheep farm there and the hills are dotted with sheep. I'm sad that we did not manage to make it to Birling Gap and East Dean. We'll have to go back someday. The scenery we did get to see was bucolic and breathtaking.

Why do people call small settlements where not many people live "Villages" in Europe.. but at home in North America we call it "living in the boonies"..lol

Villages don't have trains.. why would they.. trains stop in towns.. they need to make it worth their while.

Well I'm sure that you're right about villages not having trains, Pat, but when I lived in a village in the English West Midlands I lived in a village and about 2 miles away was another village and the train stopped midway between them twice an hour - still does in fact, I was there today - and did a fair trade.

Proper village, too. Village shop, small school, just 2 pubs, 14th century church with a nice ring of 4, ewe's milk cheese makers, thatched roofs on some houses.

Consider visiting Finchingfield in Essex, then Thaxted - beautiful villages, plus pubs and medieval church. Train Liverpool St to Braintree, then 15 minute taxi ride.

When Americans live in the "boonies" that usually means in the middle of nowhere with nothing to do. Villages, on the other hand, are usually a cluster of development with small cafes, a pub or two, cottages, a cathedral - nothing like I've ever seen in the boonies!

Sorry, but you won't find a cathedral in a village!

A village is "a small town in the country" according to Merriam-Webster; it's larger than a hamlet.

Not necessarily. Maybe in the US, but not necessarily so in the UK. Villages and hamlets are similar and either can be larger or smaller, the key difference is that a village will have a church, and a hamlet has no church.

A village also need not be in the countryside. Hampstead, for example, is known as a village within London.

A cathedral by its very definition is the bishop's official throne

A cathedra is the bishop's throne. A cathedral is the building which contains a cathedra.

A village is "a small town in the country"

Contradiction in terms. A village is smaller than a town, which is smaller than a city. A hamlet is smaller than a village. A settlement is either a village or a town or a city, it can't be more than one. But the classification is a matter of personal opinion.
I would say (at the risk of starting a "lively" discussion), a village is less than about 2,000 inhabitants, and a city more than about 200,000. But it also depends on the feel of the place.
A village rarely has a station, a town almost certainly does, and a city will probably have suburban stations as well.

And many historical places which were once the largest and most important places in the country have been overtaken in size, and though medium sized by modern standards, are still commonly called "cities", Plus, any where with a cathedral is a city (York, Lincoln, Chester, Winchester, Wells).

Some good ideas here...

http://www.beenthere-donethat.org.uk/londondaytrips.html

Thank you, Nigel.