Bristol is its own county, often known as City of Bristol. It was formerly in the county of Avon, but that no longer exists, despite almost every online form suggesting it does!
Bristol is 80 miles from Exeter by road, which would take you about an hour and 30 minutes. If you go via train it will take you about an hour an 15 minutes.
There are far too many counties in the UK to list here. If you mean 'countries' - they are England, Scotland, Wales and Northern Ireland.
Here is an example i found on the web, It lets you know...