Don't panic - I used to work as a photographer and it all depends on what you want to do with the end product. Do you want high quality prints, or are you more likely to share online?
Cameras have moved on so fast and my advice would be to go for whatever gets you what you want with the least trouble!
Compact cameras are cheap, easy to use and have good picture quality, but not such good zoom quality if you want to get close-up.
DSLRs are a total pain for the sort of family or holiday photography that most people want to do. Whilst it is nice to have some sort of greater control than a compact can offer, having to change lenses to go close or distant is a nuisance. I stopped using a DSLR even when I was working professionally as I was usually photographing film shoots and had to seize the moment when a good shot materialised - no time to faff around with lens changing.
What you need is a "bridge" camera: DSLR quality with ease of use. They have come down in price remarkably, produce excellent prints quality, give you control if you want it but have various auto functions if you are in a hurry, generally have good zooms and no lens changing needed! Look for a good "optical zoom" quality on the details - some have enormous "digital zoom" which gets in close at the expense of picture quality.
If you go out and buy "Which Digital Camera" they will review them all and usually have a section for bridge cameras.You can lso research bridge cameras online. Having chosen which one you want, then shop around online as it is a competitive market and you can often get a good price. If you find one you think looks like what you want, post on here and I will let you know what I think.
One other factor to consider is power source. I have one camera that works on ordinary AA batteries, so if I run out of power with the rechargeable AAs when I am out and about I can just buy off the shelf to keep me going.
If you are taking family photos, make sure you put the red-eye facility on when using flash, and - a little tip here! - if you still find the flash too harsh (with a too bright picture and harsh shadows) lick a cigarette paper and stick it to the flash - works like a charm!
Also think about weight - do you ant something heavy hanging round your neck?
My two bridge cameras are an Olympus (which is very old but I love it and it's an old friend of mine) and a Fuji.