Thankfully these are golden times for learning about electronics. What with the internet providing pretty much all the information you'll ever need, along with very cheap electronics modules/components/tools mass produced in china means 'cost' is no longer a limiting factor.
Anyone who says you "can't do electronics without a scope" is high. There is plenty you can do without a scope - but it depends on what these people mean by 'electronics'. To some this means opamps, filters and oscillators, to others its microcontrollers, LED's and motors. For analogue stuff, yes a scope helps a great deal, specifically with AC stuff, amps, filters etc.. but for beginners, whilst learning the basics is obviously a good way to start, it can also be boring. You don't have to spend >£100 to get started. Not even close.
I would say the best way to learn is to find a project you want to make/build, something that isn't overly complicated, but will challenge you. If this project requires some analogue stuff - then that is the opportunity to learn about them. Don't try and cover everything at once, learn different areas as and when you *need* to know about them. An Arduino is a great piece of kit, that, along with some 'shields' you can do the sorts off things that would have taken a hobbyist a lot longer a decade ago.
These days there tends to be more emphasis on software, which isn't strictly electronics, and is often why when people hear the word 'electronics' they not only think of hardware (obviously) but also, non-programmable stuff. But microcontrollers are a huge part of electronics, and can be a good starting place: You can think in terms of 'modules', with a microcontroller controlling them. Input modules (switches, buttons, sensors!) and output modules (LED's, motors, displays) can all be broken down and isolated, allowing you to see how each "module" works. These can then be hooked together to form actual useful hardware.
tl;dr: Learn as you go. Pick a fairly ambitious project, and work your way through it, step by step.