I don't think you can go wrong if you stick with one of the name brands, like Nikon, Canon, Sony, etc. Go someplace you can actually pick them up and get a feel for them. It may sound great to have a really compact camera until you are holding it and all the buttons are so close together your fingers keep hitting the wrong thing. Or the camera that looks so small and pocket friendly online is too big for the pocket and heavy as a brick. You also want to see the optical zoom in action to see if there is enough capability to suit your needs. Buying a camera is a very personal preference. What works for me won't necessarily be the best choice for you.
Personally, I love taking photos, so I have three cameras besides the iPhone. I have a very basic point and shoot that I carry if I'm going somewhere the camera could possibly get damaged. I have a super zoom camera for those times I want good photos but don't feel like carrying a lot of equipment. I have the Canon SL1 dslr, which is a smaller and lighter weight dslr, and assorted lenses, that I use when I want great photos and don't mind carrying an extra lens or two. All my cameras happen to be Canon, but other family members use Nikon and are just as happy with their results. The Canon's just feel good in my hands and the controls are well placed for how I use the camera.
My point and shoot and super zoom both use AA batteries, which comes in handy if you're taking photos where you can't charge a battery. Even with having two charged batteries for the dslr, I've had some days where both ran down and I didn't have any way to charge them until I got back to my hotel. I use rechargeable AA batteries regularly, but there's nothing like the convenience of just picking up a pack of batteries at a gas station or convenience store and continuing to shoot.