In the UK we have one of the densest and most comprehensive road networks in the world and if you are used to travelling by car you probably have no idea that they are there. With the exception of bits of Wales and Scotland you can get almost anywhere by bike without using major roads. The reason we usually want to avoid major roads is that those are the ones that carry almost all the motor traffic and cycling is much more pleasant and relaxed on quieter roads. In the southeast of England we are cursed by very high levels of motor traffic but we are blessed with an excellent network of minor roads and lanes that go everywhere and are much less busy than the routes you would use as a car driver. What's more, by bike it doesn't usually take any longer to get from A to B on the lanes than on the main roads.
We have a hierarchy of roads in the UK going from single track lanes with passing places up to motorways. In a motor vehicle you will prefer the bigger, faster roads. On a bicycle we prefer the smaller roads. I have listed the road types below along with the ways I would use them when planning a route. This is strongly biased towards the types of roads we have in the South East of England and some of these generalisations don't hold true all over the country.
Prefixed M e.g. M1, M2, etc. or with an M suffix such as A3(M). Marked in blue on maps. Bicycles not permitted. When route planning try to avoid even getting near the junctions as motor traffic will be approaching and leaving the motorway at very high speed on roundabouts that are often highly unsuitable for cycling. However do not go miles out of your way to avoid the junctions if they are your only sensible option for crossing the motorway, you will often find that with care a sensible cyclist can still negotiate the junction although it may be unpleasant.
Primary A roads
Prefixed A followed by one or two digits e.g. A3 or A31. Marked in green or red on Ordnance Survey maps, and yellow in some road atlases. Some of these are motorways in all but name and should therefore avoided. Others can be OK at some times of day. Generally avoid these roads if they show as dual carriageways on the maps or if they appear to be the only route in the area. Through town though some of these roads will only be 30 or 40 mph roads and are fine to cycle with care, albeit with more traffic than some people like.
Main A roads
Prefixed A followed by 3 or 4 digits. Marked in red on Ordnance Survey maps, and yellow in some road atlases. Usually the line denoting the road on the map is narrower than that for primary A roads. Generally usable with care. Minimise the time on these roads, unless you know them to be ok for cycling. Best used as links from one minor road to another if you prefer your cycling to be tranquil and traffic free.
You can cover the whole country on the B road network, and very pleasant it is too. These roads are marked in orange on Ordnance Survey maps, and white on some road atlases. These roads can form the bulk of your route from anywhere to anywhere else. Some can be very busy, especially in the south east in which case switch to the more minor roads.
Other minor roads
Shown as yellow or white on maps. These lanes really get you into the countryside. Some people like to use these in preference to the B roads whenever possible, as indeed I do, but it must be remembered that sometimes excessive length can be added to your journey by trying to stick to these roads.