stone circles via rail/bus

From the center of Keswick it is not a bad walk to Castelrigg. I've done it and it's a fairly sharp uphill as you get toward the circle but otherwise doable. Keswick is reachable by public transport.

Here is a link to a pretty comprehensive list of megalithic sites. I've not used it enough to see if there is a way to sort by public transportation or not.

http://www.megalithic.co.uk/index.php

The ones I've been to in Dorset and Cornwall have not seemed particularly accessible by public transport. I'd think one of the problems is that the sites are sometimes really small so it might take you longer to get there than to see - if that makes sense. The ones I've seen have been associated with either a Rick Steves or Road Scholar tour except for Stonehenge which is easily done by public transport.

If you need information on getting out to Stonehenge, I'm happy to share, just didn't want to include that if you've seen it already.

It's about a 30-40 minute walk from central Keswick up to the stone circle at Castlerigg. Like the previous poster said it's steady uphill walking up Eleventrees for the final 1/3 or so...just take your time! I would highly recommend going either early, very early in the morning or late in the afternoon. It has become too popular for its own good and can be quite crowded now, unfortunately.
If you don't want to walk you can take the local bus up there. The bus that goes along the A591 stops right at Castle Lane that leads right to the stone circle...about a 5 minute walk between hedges, cows, and sheep. You can ask the bus ticket attendant that is usually at the main bus depot in front of the Booths store in town...they will sell you a ticket and tell you where to get off...very easy bus system in the Lakes!
If you want a nice walk back DOWNHILL from the stone circle you can go back to where you got off the bus, cross the A591, go through the stile into the field, and walk toward the mountain. Just follow the footpath through the different fields and paddocks until you come to a tiny paved road. Directly across that is another stile and a path leading down into the woods where you can spot red squirrels and head back into Keswick. Or you can head to the left along the tiny road and take the path over the footbridge and up a very steep path to Walla Crag. Spectacular views over Derwentwater and Keswick if the weather is clear. Both of these walks are two of my favorites...beautiful views and classic English Lake District.

Long Meg & Daughters is about half a mile north-northeast of the Penrith railway station, as the crow flies. The TI office in town is friendly and helpful. In the high street there's a nice pub called Narrowbar that's good for lunch.

You can see the train service here https://www.thetrainline.com/train-times/penrith-to-london-euston

There are buses 7 days a week from Penrith railway station to Keswick, according to Rome2Rio. However, it looks like there are only a few buses per day so you'd want to watch the timing.

We did a stone circle-themed tour (with a rental car) a couple of years ago. Long Meg is not as well cared for as most of the other circles we visited. There was a notable amount of graffiti on the stones, and the cow pies in the pasture were gigantic -- watch your step!

Another thing to be aware of is that in Keswick, the public toilets aren't in or near the TI office. They're at the public car park, not very well signposted.

If you're going to Stonehenge, make time to not only enjoy the large visitor centre, but also to walk out to Woodhenge and Durrington Walls. Visiting those sites really adds to the experience, IMO. https://www.nationaltrust.org.uk/stonehenge-landscape/trails/durrington-walls-stonehenge-landscape-walk

Check train schedules & fares at www.nationalrail.co.uk
(For longer journeys, book specific trains about 11 weeks ahead for lower fares. This may work out cheaper than Britrail passes).

The Avebury stone circle is very accessible via train and bus.
Get train to Swindon then bus to Avebury ( the Number 49 bus runs hourly)

I second Avebury. You could make it a day trip from London.