I agree...but not everyone is interested in photographing, and I've seen some excellent phone photos. I have no idea how - I love photography, study it and take classes, and people say I'm very good for an amateur (and I've had a few sales to friends and family), and I can't take a decent iphone photo to save my life. My phone is a 5S so not the newest but not an old one. MAYBE the 6 is that much better, I will need to wait a year until my upgrade to see :)
IF you are interested in photographing your trip, and IF you aren't skilled with a phone camera (most people aren't), then get a basic digital. I carry a canon digital point and shoot (under $100) and a basic Nikon DSLR ($400) with a kit lens. The DSLR is notably better than the point and shoot, but the p&s has done well. I wouldn't recommend the DSLR unless you are truly interested in photography as an art - it is heavy and takes up room, requires a larger daybag, etc. But the point and shoot is an excellent compromise.
Shoot like crazy - I don't. I take several hundred pictures a day and many turn out to be junk. But I don't just shoot willy-nilly. Take the time to look for good angles, play with your settings if you have them and take pictures of unusual things. Another broad view of the altar? Maybe boring. A close up, or a from-the-floor angle up to the ceiling fresco? Maybe interesting. My #1 best photography advice is MOVE. Don't just stand there and shoot. A few feet to the left or right, kneel down, stand on a ledge (when safe and appropriate), whatever - the best angle is often a few feet away in another direction. Move left and that telephone pole and tour group are out of the picture. Sit on the ground, point the camera up, and the tree branches frame the bell tower instead of obscuring it. Etc. Just be creative.
Ruthless in deleting - I'm not. Every night I upload a lot of that days photos on to my ipad (leaving out the blurry ones, the duplicates, the ones where someones head popped into the frame at the last second...those are deleted all together), then go through and keep the 20-30 best from that day on my ipad, and upload them to dropbox. The rest are deleted from the ipad but live on the memory card When I get home everything but the junk goes on the computer. However, the 20 per day chosen on the road are usually the best and the ones I work with through editing to show to people, some get framed and hung in the house, etc. This gives me all my photos available somewhere, but narrows down the editing and selection to a more manageable number. 20 per day is 200 for my average 10 day trip on the ground. I usually take maybe 2000 pictures in that time. Plus if I loose the memory card, I have the ipad copies. If I loose the ipad, those best of pictures are online (dropbox or somewhere) so I've saved the essentials no matter what.