The part of the world you want to travel through has some of the best public transport systems in the world. There are countless options for travelling with trains (faster, more comfortable, more expensive) or buses (slower, less comfortable, cheaper). Not possible to list them all, but you should definitely have a look at the Seat61 website (see jcaron's answer). However, it's not likely you'll be able to buy a single ticket for your entire journey.
The fastest and most straightforward (but probably not the cheapest) option would be an Eurostar train to London and another train to Scotland from there. If you're not on a tight schedule and are willing to experiment a bit, you can search timetables for cheaper connections, either trains or buses. You can also cross the English Channel as a foot passenger on a ferry. There are countless ferry connections, from France or from the Netherlands, and you can save a lot of money by doing some research.
Maybe it's worth mentioning that you can buy a rail pass (e.g. InterRail) for the UK (or for any other country but a UK pass would be most useful), if you want to travel around. Just keep in mind that some long-distance train services in the UK (as well as in other countries) require a reservation that must be bought separately if you have a rail pass.
And the last piece of advice: don't forget your passport! EU nationals in general cannot enter the UK with their ID cards.