I wish to add a shapefile to ArcMap that displays a road network which is similar to what you may find on Google Maps. I don't require a detailed Ordnance Survey map, just the roads.
Indeed there is! Look at the Ordnance Survey Open Data website, and the product you probably want is Meridian 2. It's free to download and use for even commercial exploitation, but check the licence just in case.
It has layers other than roads, but they're in separate shapefiles, so naturally you can ignore them. It's designed for mid-scale mapping, so don't expect very high resolution. You can combine it with OS Locator if you want to do some more complex searching.
You could also try OS Vector District, but I've got a feeling the road layers are derived from Meridian 2 anyway.