Steam train rides in and around York

There are two obvious answers. Firstly, the National Railway Museum is in York, right by the station. It is also possible that there could be organised steam day trips from York station, although these will be only occasional.

Then there is the North York Moors railway, which runs steam trains across the national park from Pickering to Whitby - one of the longest preserved steam railways in the UK.

There are lots more preserved lines all across the UK, should you be visiting elsewhere.

One of the less well known options, on certain dates is the Scarborough Spa Express - run by West Coast Railways, who are better known for the Jacobite in Scotland.
On certain days they also run the Dalesman to Carlisle over the Settle Carlisle line.
Easily accessible is the Keighley and Worth Valley line at Keighley.
Also the Embsay and Bolton Abbey line just outside Skipton
Whitby to Pickering is the North York Moors Railway.
Although not normally steam there is the Wensleydale railway in Herriot country - other than on a Sunday a bit difficult to get to without a car.
Up close to Durham there is the Tanfield Railway, marking its 300th anniversary this year.
Near Leeds, so easily accessible, is the Middleton Railway, dating from the 1750's.

Sometimes steam hauled specials appear on the main rail network. You can find these on the following website - which also includes ‘classic’ diesel haul trains. https://www.railwayherald.com/railtours

For the train enthusiast there is the Sidings Hotel- 10 minutes by car from York or Poppleton stations- right beside the East Coast main line. Five converted old train cars converted into a boutique hotel and restaurant.
Anyone who has gone north from York by train will have seen it.

Another interesting little operation (open at weekends) is the Shipley Glen Tramway. It is really at Saltaire, which is on the way to Keighley and Skipton.

As well as the North York Moors Railway from Pickering to Whitby, There is also the Keighley and Worth Valley Railway that runs between Keighley and Hawaorth. It is a shorter run. (Catch a train to Leeds and then a train to Keighley).

For something completely different, there is also the Derwent Light Railway on the outskirts of York. This is open weekends And uses small diesel locos on a short line of track. It is unusual as it was a privately owned standard-gauge railway in North Yorkshire, England, and was never nationalised, remaining as a private operation all its life. It ran between Layerthorpe on the outskirts of York to Cliffe Common near Selby. For lmore information have a read here.

Since Scarborough has been mentioned (it is just a bit south of Whitby), I wanted to add the Scarborough Central Tramway: https://www.centraltramway.co.uk/

It's a funicular, not a train, but to me it captures some of the same spirit as a steam railway as it's also from Victorian times. Worth a ride if you are visiting Scarborough.

First, many thanks to everyone who listed possible heritage rides. Our problem is that we're time limited (aren't we all time limited when traveling?), so we're going to have to limit ourselves to the NYMR railway and the GWR which runs out of Broadway to Cheltenham. It's not so bad, we're doing all out travel in country via trains, just a lot of diesels and electrics!, which has to be better than autos or airplanes.

You've chosen well there. NYMR railway and the GWR Broadway to Cheltenham and both decent length rides which ensure the steam loco can really sjow off its pacces. Scenery along the two is also different.