Hi ts -
My twopennorth would be to take the train from Manchester to Penrith, then bus, or if feeling flush, taxi to Keswick for the Lake District. How long in the Lakes? I’ve been hiking up there for almost sixty years and still have plenty left to go at, so you won’t ‘do’ it all in a couple of days. But the great thing about Keswick is there is a ton of hiking straight out of the door and as I’ve mentioned before in other threads one thing you won’t want in Keswick is a car as parking is either non existent or eye wateringly expensive.
Alternatively, take the Transpennine rail service to York direct from Manchester. Again the one thing you won’t need in York is a car and if I persuaded you parking was a nightmare in Keswick, well York will see that and raise it!
That said I think I would try and pick up a car in York for the Scottish leg of your journey. The only way to get round Scotland easily and comprehensively is by car. And it means you can take in Hadrians Wall on the way to or from Edinburgh and not involve backtracking. (My thinking on this is that you return the car back in York and rail back to Manchester - some of the Transpennine trains actually terminate in the airport and I would recommend staying overnight in one of the airport hotels if you have an am or early pm flight departure). Hadrians Wall is best seen in the rugged central ‘Whin Sill’ section as it snakes over the landscape near Once Brewed and the Roman remains of Housesteads (on the wall) and Vindolanda (set back a bit from the wall) are comparatively nearby as is Sycamore Gap of ‘Robin Hood, Prince of Thieves’ fame!
I’m not sure that there’s a heck of a lot in Inverness to detain you. Culloden and Clava Cairns are nearby I guess but the town itself is OK, but nothing remarkable (apologies to the inhabitants of Inverness - I say the same about my own city!). I get the castle and Wallace monument in Stirling though. There’s not a heck of a lot in Glencoe either except a wealth of steep and tough hiking which in good weather would occupy you for days on end. Accommodation in Glencoe is limited, but you may find places in Ballachulish, Onich or definitely in and around Fort William (An Gearasdan locally - means ‘The Garrison’ in Gaelic and is what you’ll see on some road signs together with the anglicised ‘Fort William’).
Skye is very much worth the effort in my view, especially for hikes into and amongst the Black Cuillin, but be warned, this is the preserve of the hardiest of hikers and thrill seekers. If that’s you, you’ll love it but care is advised! The abrasive gabbro rock is very sticky but sharp and will slice lumps off your boot soles (and you definitely need boots in there - do some ‘You Tube research’ - I recommend the channel ‘Scotland’s Mountains/Steaming Boots’ - to see if it’s the sort of terrain that floats your boat).
If you decide there isn’t enough time to visit Skye then the whole west coast of Scotland up into Torridon, Sutherland and Assynt is scenically stunning, although habitation is fairly sparse. In general it’s hard to go wrong in Scotland as far as the scenery goes.
Hope that might have helped in some small way.