Moved to a rural area 18 years ago from London. I can't think of many cons, so will list those first:
Definitely agree with PPs about mud. This is probably uppermost in my mind because of all the rain we had earlier this year!
Also agree with PPs about distance from big towns &/or cities meaning you have to allow a whole precious day off to go there. There is a lack of choice in local shops. I don't mind paying more though, because the quality is good (especially locally produced foodstuffs) & I haven't had to use fuel to go & buy it. Our local Post Office has recently closed down & I now have to travel 25 miles to my bank.
My area has a dearth of exercise classes, at least on the days I could attend one. Shift work doesn't help with this, but it's not like there's 3 large gyms with a panoply of classes to choose from just a stone's throw away (which there was in London, even 20 years ago).
Pros:
The space, the beautiful countryside, the views, the wildlife, the farm animals, gorgeous country walks, birdsong, stars at night, amazing sunsets, beaches a short drive away.
Lack of traffic & traffic jams. Always being able to find a parking space & free parking.
Friends from a wider demographic: older, younger, incomers, locals. In London my friends were more likely to be my age & have children the same age as mine.
The schools here are better as they are more genuinely comprehensive. In London parents would move to get into catchment areas of favoured schools and some secondary schools took a % who'd passed an entrance exam. That doesn't happen here.
Peace, I find it much more peaceful here.
I did take DCs here there & everywhere (apart from school, there were free school buses for that), but once DC1 had passed his driving test, he could help out, then DC2 - there were 2 to help out with lifts & so on. DC3 got a moped at 16, which also helped.