How to Select a Digital Camera

I've been an avid photographer most of my life. I do the majority of my photography when I'm travelling: there's a category of cameras for that. But after 33 years behind a camera I'm pretty picky about the details: I can't just buy the most popular travel camera / superzoom. Your requirements aren't likely to be the same as mine, but walking through my selection process should help you think about what features you want and why.

When I first began writing this, I started with an explanation of what's changed since the days of film photography. But that's not terribly helpful. I do need to cover one thing though: focal length equivalency. The size of the sensor in digital cameras has remained a moving target, so the real focal length of a digital camera lens doesn't convey much information to buyers: instead, the numbers cameras advertise are "35mm equivalent." So if a point-and-shoot digital camera tells you it has a 28-110mm lens, it's flat-out lying: the lens actually has a much, much shorter focal length because the image sensor is tiny compared to a 35mm frame. But this is a helpful lie: it gives you a common point of comparison, giving you a good idea of the field of view the lens has.

I'm not suggesting you end up with the camera I ended up with: it was a pretty expensive proposition, and there are plenty cheaper out there that still produce excellent photographs. Maybe your cellphone is "good enough" (this is likeliest if you own an Apple iPhone 4, 5, or 6, as they have superb cameras - but no useful zoom and no manual mode).

viewfinder (preferably digital)
I want this because if I only have the screen on the back, I have to mess about with my reading glasses: the viewfinder eyepieces all(?) allow diopter adjustment so I can set it and use it without my glasses. I prefer a digital viewfinder as I'm not buying SLRs and - without the mirror - this is a more accurate reflection of the picture you're going to get than an optical viewfinder provides.
fully articulated screen
I have a lot of reasons for wanting this: shooting over the heads of crowds, but also shooting in churches where I'm not allowed to use a tripod. I can lay the camera on the ground and shoot the ceiling at slow exposures - and still see the screen. Try that with an unarticulated screen. There are several types of camera screen articulation: merely tilting is inadequate for my usage. Even the screens that tilt to 180 degrees aren't enough: that's great for selfies, but there's a reason the type I like is called "fully articulated."
voice memos
Both the Canon SX10IS and the Nikon 5400 had this feature that apparently not many people used since it's completely died out in the last seven years. I used it heavily, and I don't think the new camera has it. When you visit 10 - or even 20 - sites in one day, it's incredibly helpful to be able to add a voice memo that says "now entering San Carlo Alle Quattro Fontane" to differentiate it from the last several places you visited. One common solution is to shoot the sign outside the building you're entering: this is great, but only works if there IS a sign.
AA batteries, not battery packs
Yeah, sorry, that's just not happening anymore. I managed it in 2005 and 2009, but it's extremely uncommon now.
~24mm equivalent focal length on the low end
I do a lot of architectural stuff. 28mm is the absolute minimum, and on the Canon SX10IS it's been okay, but there have been times it wasn't low enough. 21mm would be ideal.
~200mm equivalent on the top end
I do a lot of architecture, but I also want to be able to photograph the details. Please note: "digital zoom" is NOT a substitute for optical zoom. "Digital zoom" merely means "we're going to programmatically increase the size of a portion of the picture and then use interpolation to try to fill in the details we lost." It doesn't work well.
3cm close focus
Very common, but not an absolute given. I love doing macro photography. I think close focus gets less common as the sensor size goes up: pay more for your camera, lose a feature - sigh.
shutter priority
This is my preferred shooting mode.
full manual mode
Cameras are smart these days, but sometimes they're really dumb or you're trying to achieve something different than what they think you want.
filter mount on lens
Not critical, but desirable: better you ruin a $50 filter than the front lens on your $800 camera. As well as protection, polarizing filters are a truly wonderful thing.

Something to consider is the size of the sensor. The common size for most point-and-shoots is 6.2x4.5mm. The dream of those of us who remember the film photography days is a 36x24mm (the same size as the 35mm film frame). The problem is - the sensor is the most expensive thing in the camera. And the bigger the sensor is, the bigger the lens glass has to be to cover it - and the lens is the second most expensive part of the camera. It's not all bad: small sensors are also what's made superzoom's possible. Especially since in broad daylight, any decent small sensor point-and-shoot will do at least as good a job as the best old 35mm film camera ...

Things I don't worry about much that you maybe should:

sensor size
A large sensor is highly desirable for picture quality, particularly in low light condition, but it has some nasty side-effects: 1) the cost of the camera increases exponentially, 2) the cost of the lens(es) increases a lot, and 3) zoom range of lenses is generally cut down.
I've actually grown to dislike the SLR: a good digital viewfinder gives you more information, particularly about the exposure as it's going to be recorded. And the lack of a mirror means you get most of a stop back because camera bounce ceases to be a problem. Not to mention the space savings.
widest aperture
I take the lens I get and deal with the fact that it's only f2.8 or f3.5. Some people find this unacceptable and consider f2.0 a minimum. If you're one of them, you can kiss your zoom lens goodbye: they don't support f-stops like that, not without breaking the bank.
I haven't started shooting video yet. A lot of cameras, even ones meant primarily for still photography, can shoot 1080p video out of the box. But you want different controls and different behaviour that you'll need to look for. And if you want 4K video, that's a lot less common.