The basic thing you are stumbling upon is the difference between layer CSRCRS and project CRS.
The OpenstreetmapOpenStreetMap 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 OpenstreetmapOpenStreetMap Website.
And On-the-fly-projection
should be activated.