Cycling in Europe

Taryn, Switzerland is very cycle-friendly. There are lots of cycle routes, both national and local, which allow you to traverse the whole country without being bothered by "nasty smelly noisy fuel burning metal boxes".
Start your research with the official website here: http://www.veloland.ch/en/veloland.html

Cycle routes in the country are often off-road, and shared with walkers and horses. For example, in a valley, the road may go along one side of the river, and the cycle and footpath along the other side of the river.
Cycle-friendly also applies in towns. One-way streets and pedestrian areas are designed to stop cars going through the middle of towns, but usually bikes are allowed. Most urban one-way streets allow bicycles in both directions, so cyclists can take the direct route through the centre, but cars can't.
See this road sign here: http://blog.veloplus.ch/wp-content/uploads/2016/01/verkehrsregeln_teaser.jpg
This is a "no entry" at the wrong end of a one-way street, but "gestattet" = permitted, i.e. bikes are allowed to go down the street. At the other end of the street you get a sign like this: http://www.med-etc.com/v/verkehrszeichen-CH/hinweissignal005-einbahnstrasse-geradeaus-m-velogegenverkehr.png
One way street (blue sign with arrow), but a warning triangle underneath saying it is 2-way for cycle.

Trains all allow bikes on board. And, supermarkets in urban areas, including my local supermarket, do not have car parks. But, they are usually located near public transport, and do have bike parking racks.