There are a lot of cartridge razor blade subscription services spamming Facebook at the moment, and I've been happy with my shaving routine for years at this point so I figured I'd post so that I can link people here in future. Whilst some people like straight edge razors for the cool factor, some people have never known anything but cartridge razor and some are perfectly happy with disposables, I like the middle ground of a traditional double-edged safety razor. They've been around for a hundred years plus, are cheap and effective, and despite the best efforts of razor manufacturers to sell everyone overpriced toss with multiple blades, aloe vera strips and vibrating handles, etc, they remain popular today.
These are all Amazon links for convenience, but that isn't necessarily the best place to buy and there's usually plenty of everything on eBay.
Ideally a razor should be heavy enough that you don't need to use any pressure to draw it across your skin, and the Futur has that kind of satisfying heft. It's also adjustable, which means you can control the angle the blade is at. I generally leave mine on 2, which gives a close enough shave for my taste. Anyway, they're a bit pricey but are easy to change blades in and will last a lifetime.
A cheaper option for trying more traditional shaving is the Wilkinson Sword Classic, which is well made and the angle it holds blades at isn't aggressive. Or, if you like collecting stuff, the adjustable razor Bond uses in Goldfinger (1964) is a Gillette Slim. I've got one that's been re-plated, but it's a bit of a novelty because I do really like the Futur. Apart from eBay you can try http://www.executive-shaving.co.uk where I thieved the image from.
Around ten quid for 100 blades, which will last a long time even if you like to change them regularly. It's difficult to find double-edged razor blades in supermarkets because there's very little profit margin on them, and when you do they tend to either be own brand or ones like Derby that aren't very sharp and suit razors that don't allow you to adjust the blade angle. It's better to shop online and it's also definitely worth trying more than one type with a mixed sampler pack on eBay and finding ones you like. In a non-adjustable razor I used BIC brand blades for a long time, but having found Astra Platinums I've stuck with them because they're sharp without any skin irritation for me personally. The ones that come with the Wilkinson Sword Classic aren't bad either, and I found them pretty much equivalent to BIC brand ones.
This does contain unpronounceable chemicals, so isn't much better than shaving soap in that respect, but it's a couple of quid, lasts ages and whips up easily unlike many shaving creams and some soaps.
Using oil before soap makes it much harder to accidentally cut yourself, and it doesn't have to be a anything fancier than olive oil (the Romans, etc, were big on it and apparently it's very good for skin). Some people even use it instead of soap and stand their razors in it to keep them sharp.
People will try to sell you badger hair brushes because they're soft, where soft supposedly equals quality and gets more air into the soap, blah blah blah, but to cut a long story short I like badgers. I use a coarser organic bristle one (if the source isn't mentioned, it's probably from boars) but the one above is synthetic and just as good if animal products bother you. It's worth getting a stand to hang a brush from rather than put it on its base, otherwise the bristles are likely to come out even if plastic ones won't rot.
Once you've steered away from companies paying celebs to hawk cartridge systems for a premium, there are lots of other (often also quite expensive) gimmicks to look into. For instance I've tried shavettes, which are like a straight razor but usually take half a double-edged blade, which are more intended for barbers tidying up the edge of a trim. Actual cutthroat razors require care and maintenance, plus more skill than I'd trust myself to have regularly whilst half-asleep in the morning. And I'm not interested in pampering myself with different soaps and other products, although fair play if you like a bit of that.
May 2019 update:
It appears the patent on the Futur has expired, which means that clones are now readily available on eBay for about £8. Some of the discussion dating back to 2013 suggests there were quality control issues at the time, but I can't see any issues with the Ming Shi 2000S "sharp-adjustable shaver" I got as a spare from a seller drop-shipping from the UK. I'm also not seeing the tightness in the adjustment mechanism that those early buyers refer to, or any finish defects – it's possible there are a few companies in China doing these and that Ming Shi and QShave, who also get named in listings, are just two selling to Western markets. Or that it's all one company and they've just refined the process. They're slightly lighter than originals and with slightly more defined tapers in the handles, but overall excellent and worth a try (especially at that price).
Reference to 1983 patent:
2017 2000S review: