I agree completely. OP says they have a cellphone camera, and want something better, but not hobby or pro. "Family, friends, projects, that actually look good." This screams point and shoot. One could make an argument for the "interchangeable lens cameras" or whatever they call the range between the p&s cameras and DSLRs this month, but I'd go with the point & shoot. Based on my research (do your own, visit your library and check Consumer Reports and some photography magazines) Canon and Nikon do cost more, but you're paying for quality. To me it's worth it; YMMV.
There's a lot of good, accurate information posted here about DSLRs, but they aren't what the OP wants. If the OP already understands setting IOS or using auto-ISO, then sure, go for DSLR. If not, get a point and shoot.
I bought a Canon Powershot A590 a couple of years ago, and it was great (and still is) until I started shooting roller derby, a sport that is high speed and often played under low light in facilities that were not meant to be sports arenas, so it wasn't up to the task, so I bought a DSLR. I've racked up 26,000 photos this year on my Canon 60D, but this is not the camera for Soulskill.
What the OP wants is a point and shoot with some amount of zoom and some advanced shooting modes (Program, Time/Shutter, Aperture, Manual) so they can learn about them if the standard modes don't meet their needs. Get one that's small enough that you'll actually carry it (mine fits in my jacket pocket), and one that's comfortable in your hand (yes, go to stores and handle them). Get maybe 10-12 megapixels; don't pay for higher (unless that's all that's on the current market). Pay some attention to the memory system -- does it use SD/SX cards, flash cards, or Sony power sticks? You won't be happy with some weird proprietary memory media. I'd suggest starting to look at the Canon A590 (or whatever the current equivalent is) and adjust based on desires and constraints.
Also, download the photos off of your camera. If it takes too long with the supplied cable, you can get a card reader (mine was $12) to convert it into a USB drive and download ~10 times faster. But get the photos off your camera so you don't lose them, and you can actually look at them, learn from them, enjoy them, and take more. If you want to improve your photos, take lots of them, and get a book such as Digital Photography for Dummies (or reasonable facsimile thereof).
The original poster doesn't want to shoot RAW; he just wants to take easy photos of family and stuff. He doesn't even want to get to the hobbyist level. I agree that SLRs are better, I've extremely happy with mine, but they aren't what Soulskill wants.