Is the compact restriction, based on budget? or size you want to carry.
If you really want to get into photography, you want a camera that gives you some manual control.
If you think you would want to get into some serious editing after taking, you want RAW capability.
This can often improve a photo that takes on the average side, by giving you the most control to tweak.
(Until I learnt how to use my camera, this saved a lot of my first photos).
www.cameradecision.com put in a few model numbers here, and you can compare features quite easily, some sites better for reviews, but this is a good way to directly compare a few you like quickly for the features and basics.
Also, head to Flickr and type the model number in search. You can see the quality of the images the camera will take, find the one that you like the most with kind of shots you want to take, the average look of images is what you will shoot, and when you see some that really stand out, you'll see what the camera is capable of when you learn how to use it. Also, look at sample images on manufacturer sites.
The above is part of what I did to research my dslr.
If you think you will get into photography more, it might be worth looking at small body mirrorless systems and micro four thirds systems, which minus lens don't weigh too much more than the compact you mentioned and not much difference in size either, but they will have the capability of growing to your style with an extra lens or two (which are also smaller and lighter than full blown dslrs) e.g. my dad was after an all purpose like you, and he is happy he went with an olympus OMD with 2 lenses which he just got back from a month around europe with. you could possibly find a second hand one of these in good condition somewhere.
Panasonic make good cameras, and looking at flickr, the one you mentioned is capable of some good shots. If you do get into photography the way some of us have on here, you will one day want something more, but a good compact is always a good grab n go camera as well. Looks like good manual control and RAW capability so this or newer models of it will work fine. Just be wary of difference between optical zoom and digital zoom, I haven't looked in detail, but some "big zooms" are part lens, and then digital takes over past a certain point, and the quality isn't the same in a lot of cases.
Whatever path you take, you will be glad to have stepped up from an iPhone, when you learn how to use it.