In the Lake District- examples of things to do that are a bit unusual, in late April and May this year-
This last weekend just gone- Dalemain Marmalade Festival (near Ullswater)- a service bus, 509, stops there, also a festival shuttle bus from Penrith, while there go and see the Dacre Bears and (on the Sunday, yesterday) a Classic Car event
25 April to 4 May- Ulverston walk fest- 33 walks over 10 days (and visit Stan Laurel's birthplace and do the squirrel benches town trail)
28 April to 4 May- The Settle/Carlisle Walking and Music Festival (at various towns along the line)
2 and 3 May- Leyland National 54 bus festival- at Whitehaven- a weekend of events for the family centred around a vintage bus event with free bus rides around the harbour, including a fun fair, etc etc, around the Town Harbour
2-4 May- Muncaster Sausage Festival (at Muncaster Castle)
9 May- Fell Pony Stallion and Colt Show (at Dalemain Mansion)
9 May- Eskdale Herdwick Tup Fair (at Boot, Eskdale, use the Ravenglass and Eskdale Railway)_
9 May- National Mills Heritage Open Day- at Beetham- #755 bus from Windermere
9-10 May- Brougham Hall Wartime Weekend- Brougham Hall, Penrith
16 May- Langwathby May Day (a station on the Settle/Carlisle railway)
16-17 May Brass at the Castle (Brass Band Festival at Brougham Hall- use buses from Penrith to Ullswater
16 May The Halle Orchestra- Dvoraks 8th Symphony- Kendal Leisure Centre
16 May- Royal Northern Sinfonia- Haydn's Creation- Sands Centre, Carlisle
22-27 May Buddhist Spring Festival at Conishead Priory, Ulverston
22-24 May Bowness Bay Brewing Beer Festival at Kendal
23-31 May- Medieval Muncaster- festival of Medieval events at Muncaster Castle
29 May- 6 June- Ulverston Music Festival
30-31 May- Country Fest at Westmorland County Showground, Crooklands, Kendal
31 May- South Cumbria Fell Pony Show at Westmorland County Showground
Some slightly out of the usual places to visit on public transport-
Threlkeld Quarry Museum (Quarrying was once a major industry in the Lake District)
The Senhouse Roman Museum at Maryport (the largest collection of Roman Altars in the UK, a significant general Roman museum on the site of an old Roman Fort)
The Derwent Pencil Museum at Keswick (once one of Keswick's major industries). It's quite good that they give you a quiz sheet as you enter to keep you engaged with the exhibits. Also they have a sale on (25 to 30% off in the museum shop)- you could spend a lot of money on pencils and artists materials!
Lakeland Motor Museum at Backbarrow (bus 6 from Windermere, or steamer to Lakeside, then the steam railway)
Holker Hall and Gardens - 15 minutes walk from Cark and Cartmel Station.
Stott Park Bobbin Mill (another major local industry)- short walk from Lakeside steamer pier
I'm never sure what living like a local means, but-
I live in an old mining town, where we had far too many mining disasters. With little forward notice they are each solemnly commemorated- we had two this last Friday.
On a Saturday morning you might go to the local park run (every week there are visitors present), then join me in maintaining our local Community Garden (which has won Green Flag awards, on the site of an old cleared settlement). In Summer we also meet on a Tuesday evening.
On many Sunday mornings there are often community litter picks around the town.