There's one magic sign that
might help you with hitch-hiking, especially up and down the motorway network. A "garage plate"[1] plate is often seen with people on the approach to motorway sliproads. I think this conveys some sort of image of "official vehicle-less-ness" and means that these people are more likely to be helped out by people who might also work in similar areas of the vehicle couriering industry.
However, I don't know how much this would help you. The places you see these people necessitate the picker-upperers stopping on the sliproads down the motorway (I'm certain this is illegal, although hardly uncommon to see) for the hitcher to run down to do their negotiation (assuming they haven't already informed the driver with the cardboard sign saying "Edinburgh", or whatever). You may also have to get a decent back-story to justify this tactic, and the number-plate isn't going to be easy to get in the first place.
However, I do believe this method has been used successfully by someone of my loose acquaintance.
Similarly, walking with a petrol can works wonders (on a busy road, maybe only the hundredth car to pass by stopped, but that was within a couple of minutes of getting onto it, and others may have stopped if not so busy), except that obviously it'd normally only be an offer of a lift to the nearest petrol station or (if they don't mind the inevitable smell of petrol) back to your vehicle from that, so probably would upset someone if you asked for long-distance travel on that basis!
All in all, I must admit that I have never picked up a hitch-hiker. Perhaps the closest is that of offering a lift to someone I know who happens to be walking along the road I'm driving, as it is in the reverse. I believe I'm fairly representative of the British public (polite, and all that, at least outside of the inner-city environment and in among the masses, but when in ones own car tends not to let strangers into it on whims) but there's an awful lot of us on the roads and there are going to be outliers (a few very accommodating ones, plus very a very small number of scary ones, both of whom might be eager to stop, but for different reasons). But then I see very few HHs (maybe a self-fulfilling truth, if there aren't enough to be seen doing it, so those that are left don't see many car drivers that help.
For travelling around the UK, I'm really unsure as to what method of travel would work well.
I last used a train a couple of months back: Return journey cost a little bit more than the round-trip of petrol would have, a single-way journey would have cost more than the return-ticket! And then there's the inconvenience of having to find a handy station for each point on your journey, which is probably Ok if limiting to major centres of population, for the most part (thank you, Dr Beeching...) but apart from certain scenic areas visible from the lines themselves generally misses out most of the interesting landscapes. Plus often waiting times, inconvenience, expensive and/or inedible food, blah-de-blah.
Bicycle travel is better than it has been, with National Cycle Network routes set up all over the place (a lot on disused train-track beds... so... Thank you Dr Beeching!?!), although as a long-time cyclist I can tell you that UK motorists aren't anywhere near as friendly towards two-wheeled people (both motored and un-motored, but for different reasons) as their continental brethren. Plus we have hills (especially where it is most interesting to be) and wind-swept flat-lands (especially where you're likely to want to ride into the wind!) Nothing like the Alpine foothills or Netherlands reclaimed plains, respectively, but still significant barriers to the tourist who isn't used to it all. (And even one who is.)
There are a number of Long Distance Paths which are good for people who are more experienced at (non-hitch) hiking. But not for the novice, and you really need to plan your stops if you aren't confident/blasé about good weather and the possibility of camping out (noting that you really don't get automatic right to camp out on common land in England, like you do in Scotland).
Coach-travel, good for between major destinations, can usually be augmented with busses for within a given locality. But can be a crap-shoot.
On the whole, though, for true nation-wide travel, Car is King, and preferably your own. And probably cheaper in the long-term (past the initial investment in buying/hiring and associated insurance), despite the high fuel costs. And none of the above (excepting cycling/either type hiking, once you've got the equipment) is free. Right now I have difficulty imagining I could be a rep for the English Tourist Board (or indeed Welsh/Scottish/N.Irish ones), however. There are great bits of country (far too many to see everything of, in a short trip, as I'm sure you can say the same of your own land) but I'd have great difficulty giving you definitive advice on how to see it all.
Finally, a quick tale about someone I know of who
did hitch-hike. A 'furriner', he came into the UK by plane to one of the London airports, joined the YHA (Youth Hostel Association) and then hitched up to Edinburgh. Opened the handbook and found
no Hostels in Scotland except one at the north end of the Pennine Way. Hitch-hiked back there (where we were, coincidentally, and hence how we know of him) and grumbled about it... He didn;'t realise that the YHA was England and Wales only, and that there were plenty of hostels in Scotland if he got himself an S[cotish]YHA handbook.
I don't think any of what I just said helped, but FYI, anyway.
[1] A pressed-metal car number-plate with white background, red letters and border, often a rubber 'hanging strap'. Also an archaic sequence of registration letters. But sometimes I've seen a regular one (black on white/yellow) used for similar purpose... They might well have been pinched off actual cars.