What is the best way to have great (not professional) pictures to remember our trip by, with our being cumbersome?
Depends on your photography skills and interest. A lot of people like iphone photos and say they do reasonably well. Go around your town trying to take fairly representative pictures. Inside and outside. This helps you practice and lets you see how your phone handles different situations. If the camera isn't cutting it for pictures of your yard and the church down the street, it won't be any better for the Alps and Notre Dame. There are also photo apps to help, this is an area for more research if you want to use the iphone camera.
To me, even a fairly basic pocket camera (under $300) takes significantly better photographs than a phone. Especially in low or variable light. It also zooms easier and has adjustable settings so you can tweak some things.
I use a basic DSLR but this is big and annoyingly cumbersome. I wouldn't suggest that unless you are specifically interested in photography. If you have money to burn there are now small cameras with almost all of the functionality of a SLR and multiple lens choices. I want one but we are talking well over $1000...
So my suggestion is research good point-and-shoot cameras.
If great pictures are a priority use a high quality digital camera. Why? Because the lenses in phones are too small to take really high quality photos. But you can take good pictures with a phone, particularly in highly lit conditions and especially if you don't intend to print then larger than 8 x 10 or so. What doesn't exist (please tell me if I'm wrong), but I'd like to own is a camera modified to double as a phone, rather than a phone modified to double as a camera.
But great versus good is subjective. Take some phone pictures at home inside, outside, and at night. If you're happy with the results, don't bring a camera. If you aren't, you too will be schlepping around a camera and a phone.
One advantage to a phone is the ability to easily send or post photos online. My work around is a microchip/chip/usb converter that lets me use phone or tablet to post photos from camera to web. Of course I must either take the chip out of the phone or attatch the two by cord. Alternatively, many newer cameras will transmit photos to tab, tablet, or phone via Wifi or Bluetooth. Mine does this but I found a physical connection faster and easier.
This past summer I took both a small digital camera and my iPhone 6S to France and Spain. Never used the camera. The iPhone was just too convenient and the pictures were good, in some cases very good. I have to note that as I age I find myself enjoying the sights around me more without viewing them through a photographic device. In other words, I take fewer pictures.
I think Alan's point is very important. It's indisputable that you can get a decent digital camera, not too costly, that will take better low-light photos than your phone camera will. But that doesn't matter if you don't want to carry the second electronic device around with you and aren't willing to leave the phone back at the hotel.
I decided for myself that if I didn't own and use a camera at home, it made no sense to buy one to take to Europe when I could take photos with my phone.
I use my photos as painting references. Occasionally I market a photo as well though I am not a professional quality photographer. So the camera matters to me. But even the little camera I take (not SLR but with SLR apature and stop adjustments plus a really good lens) is probably not worth lugging about with you unless you have a real need for higher quality photos.
My husband's creative outlet is photography so he too carries a real camera.
Can I take this moment to complain about people who spend thier museum time taking pictures of paintings or worse yet selfies of themselves with paintings.
Try your phone first and take a camera only if you don't like the phone results.
I am reposting, but it is applicable:
Get the latest iPhone. Learn to use it. Done.
I am totally serious. I have blown up stuff for billboards from an iPhone 5. You would be surprised with what you can do if you spend 30 mins to 1 hour reading about it.
And besides being your camera it is your phone/internet connection, translator, travel guide, etc.
I think some people have provided some good suggestions but until you are happy with your iPhone shots I wonder if you will be happy with pics taken with a (high-end?) point-and-shoot camera. And keep in mind that your iPhone, at least 4S and up, IS a good point and shoot camera. I have the Canon G10 & G11, both p&s's, and while they are incredible on a tripod it takes A LOT of effort to get them to consistently produce what I get off of my iPhone 5 & 6S and my MKIII without a tripod.
Also, no matter what p&s camera you buy, your iPhone will still be easier to carry, easier to get out, and easier to get a shot off from.
I love my MKIII and my 5 & 6S. They are incredible. If you are willing to go big, get a decent full-frame DSLR. If not, go iPhone, get to know it, take a lot of pics in the run-up to the trip, LOOK AT those photos, and then take it on the road. Your iPhone photos will look much better than the poorly frame but crystal clear photos of the rest. And they will be pretty crystal clear to boot ;). My $0.02.
Also, esp. for blog use, use the iPhone's panoramic mode every now and then to 'really capture' your environment.
Here is a quick place to look for some quick tips:
http://iphonephotographyschool.com/quick-tips/
Also check out 'golden ratio' and 'rule of thirds'.
Full disclosure: I use a Canon MKIII, carry a g10 backup, an iPhone 6S, with an iPhone 5 backup. Heavy (esp. with the tripod I carry to CRAZY places).
I totally agree with the suggestion to take your iPhone out for a spin at home to see how it works for you.
I love my iPhone, and I am constantly amazed at how capable its camera is. It works great when you have decent light, your subjects aren't moving too fast, and the focal length works for you. Specifically,
Decent light: I have had very good results with my iPhone in museums and interiors. In fact, I often prefer the unobtrusiveness of my iPhone in museums. I was also very pleased with my iPhone in places like the Pantheon, which isn't an especially bright space. On the other hand, night time shots are still a bit more than the iPhone can handle comfortably.
Subjects not moving: Usually, this isn't much of an issue for standard travel photos. If you want to take pictures of galloping horses or car races, though, I would stick with a traditional camera.
Focal length: The focal length of my iPhone 6 is roughly equivalent to 28mm or so on a full frame camera. It's a versatile length for buildings and interiors and such, but it's a little wide for my taste. I find it especially challenging for portraits. By the time I get close enough to fill the frame, my subjects' noses get pretty distorted. (I am very much looking forward to the dual lens on the new iPhones.) Also, if you like to take pictures of details on distant buildings, the iPhone is not a good choice. The wide and fixed focal length is probably the main reason i won't rely solely on the iPhone for travel.
On the other hand, I almost always reach for my iPhone when I want to do videos or panoramas. It's just so darned easy.
I always travel with a larger camera (micro 4/3 these days) in addition to my beloved iPhone. Like a previous poster, I take photographs as references for professional paintings, and there are simply too many circumstances where I need to shoot outside the iPhone's envelope.