We took our dogs on a trip on the railway which is a short but really enjoyable ride - our young nephew loved it and I think the dogs enjoyed the steam train too so a great family activity.
The train is good but I think that the best bit is that when the train arrives at Lakeside it links up with the boat trips on Windermere and you can take a long trip to Ambleside or Bowness (and some other destinations) which is a really fun dog friendly day out - it does seem expensive but you save on having to find parking in the busy towns and it's a lovely way to see the beautiful Lake District scenes so I think it's worth it (and have a drink on the boat too if you're that way inclined). It's quite open though so take a coat if it's even remotely chilly so you can enjoy the lovely views with your pooch and not worry about being chilly!