Looking only at the legal aspects of this - and not at the suitability or recommendation of any particular walk.
In England and Wales, you can walk on any public right of way, or on any open access land.
Public Rights of Way include footpaths, bridleways, and byways. As a walker, you can go on any of those. You can get information on all of those from the Ordnance Survey, who've got a useful blog post giving the differences in more detail. You can buy maps from them for the areas you're interested in, both digitally and in print. The print versions will also be available in common walking areas in local outdoor shops, newsagents and the like, as well as in bookstores (Stanfords in Covent Garden is particularly well known for maps and travel guides).
You can also see OS maps for free via Bing Maps - the key is here. But mobile reception in the hills can be sketchy, and it'll be best to take an offline map of some description.
Rights of way also include most roads. And, of course, you can walk on most roads - with the exception of Motorways, and any other roads where it's explicitly banned.
As a cyclist, you're not allowed on footpaths. But you're allowed on bridleways and byways, cycle paths and most roads.
Open Access Land includes land above 600m, moors, heaths, Forestry Commission land and more. The Ramblers Association describe what is now called the Right to Roam on this land. Broadly speaking, you can walk anywhere on it, and enjoy limited forms of outdoor recreation (having a picnic is ok, swimming, cycling, camping, playing an organised game, lighting a fire are all banned - amongst many other things), Ordnance Survey maps show this land in a pale yellow colour, and Natural England have a definitive online map.
As a cyclist, you don't get free rein to go anywhere on Open Access Land - you've got to stick to bridleways, byways and cycle paths.