The basic thing you are stumbling upon is the difference between **layer** CSR and **project** CRS.

The Openstreetmap data is in lat/long degrees, so that layer should have the CRS EPSG:4326. You can set it with right-click on the layer `-> Set CRS for layer`.

The project CRS can be different, and should be in your case. You can change tthat with `Settings -> Project Settings`.

EPSG:3857 should be the projection you are used to when looking at the Openstreetmap Website.

And `On-the-fly-projection` should be activated.