Given that (going by your previous posts) you use a brush and soap or cream, you've already made a big change to the way you shave and should notice an improvement in comfort and skin quality. If using a non-cartridge razor is a step too far for you then why change? If you're prepared or a learning curve and won't get disheartened or wound-up if the first few shaves you have aren't as close as you get with the cartridge, then a DE is probably the right choice. Modern SE razors are few and far between, and I doubt you'll find any in a bricks and mortar shop even in London. In fact, online is likely to provide a better range of DE options at more competitive prices than somewhere in central London.
As far as which model to buy, it all depends on your budget. I'm assuming that, since you were going to hit central London, you're prepared to spend a bit of money for something decent. This means you shouldn't be tempted by cheap but iffy build-quality stuff like Parkers, Weishis and Timors. Next consideration is new or vintage? Given that you're obviously scared of cutting yourself, you might feel happier with something mild. In the vintage line, the Gillette Tech springs to mind, but I'd suggest spending a bit more for something English-made and TTO like a Gillette HD rocket/UK aristocrat. They're comparatively easy to use, very well-made and stylish. If you want new and mild(ish), I'd recommend the Edwin Jagger DE89/Muhle R89 range. Some don't like them (Helveticum and dodgy spring to mind), but most find them effective and fairly gentle, and they are finished to a very good standard. Otherwise, I'd suggest the Merkur 34c, which is slightly more aggressive but is pretty much the Ford Mondeo of razors.
What you've got to bear in mind is that what shaving enthusiasts call "DE razors" were marketed as "safety razors" for a reason. You've got to be either exceptionally heavy-handed, stupid or both to do yourself appreciable damage. Attention to consistent blade angle and not applying any pressure other than the weight of the razor are the most important factors, as, unlike your Mach 3, a DE razor is much heavier and has no pivoting head or other mechanism to regulate how the blade meets your face.
Missed your query on blades - there is a degree of trial and error involved as what suits one person won't suit another. Really, the best advice is to get yourself a couple of packs of a decent, middle-of-the-road blade; something like Astra Superior Platinums or Personnas. Once you develop a decent technique with the razor (you can judge this if you're getting consistently decent shaves) you can buy a trial pack with many different types of blade. Most of us regard this as being an interesting experience, but, if you really don't want to bother, then you'd be best off sticking with your Mach 3.