Haven’t been to Bordeaux yet (apart from driving to the airport to drop off some friends - not much wine there!). But recently returned from right bank Bordeaux, where we stayed in an old winemakers house on a vineyard near st emilion. I would definitely recommend visiting st emilion town for half a day if you go to Bordeaux, it’s very small, easy to walk everywhere - but very pretty and with good restaurants and wine shops. There are a lot of wine shops! It’s a bit touristy, but if you go mid week, slightly off season, it’s probably not too bad. We stayed near there for a week in July and Sunday was very crowded, but on Wednesday morning it was much quieter. Lots of good vineyard/winery visits to do round there too, and close to pomerol.
Also stayed in an apartment in beaune last year which was really good also. It’s another small city, probably around the size of Harrogate, somewhere like that. Good wine shops and restaurants again, probably slightly fewer than st emilion. The best thing about burgundy for me was how close everything on the Côte d’Or is. Beaune is pretty much in the centre, so if you have a car you can go north to cote de nuits one day, and south to the Côte de beaune another day. There’s a designated wine route which takes you through all the key villages. The highlight for me was a lunch at domaine Leflaive in puligny Montrachet, which included a tour of the winery with Olivier leflaive himself, who is really interesting and entertaining.