Trip suggestions

The Lake District is beautiful, in my opinion.

Chester and Conwy are good choices. I haven't been to Liverpool. Another good choice would be Durham, between York and Edinburgh. Hadrian's Wall is served by a public bus that runs its length, at least in summer.

You'll want more than a day trip to Conwy.
It's a beautiful place beside the water, with a medieval castle, town wall, and buildings.
As long as you're taking the trouble to go there by train, I'd check into a hotel and stay 2 to 3 nights.
There are a couple of great fish and chips shops, plus our hotel had an excellent restaurant.

You say, "So far, we are going to Bath first,"......You can take a coach (bus) directly from Heathrow airport to Bath. Cheaper than the train and the coach has comfy, clean seats, WiFi, and gets into the coach/train station in Bath walking distance to the center of town.

It's easy to get to Stratford-Upon-Avon from Bath by train.
I'll leave directions to our train experts here.
One route goes through Oxford. I have no idea if you're interested in a night or two there.
Close to Stratford-Upon-Avon is Warwick Castle and the town of Warwick; both worth a look.

Dick is right......Going up the East Coast train line, York and Durham are "must sees" in my book. Continue north to Edinburgh and stay a couple of nights before heading to Glasgow.
I would spend 3 nights in York.