Probably the top two worries I've dealt with so far have been about belly buttons and diaper rash.

BELLY BUTTONS

One thing I forgot to ask the hospital about for my daughter, was the care of her umbilical cord stump.  All I knew was that it's supposed to dry up, and fall off.  That little stump nearly gave me a panic attack when I noticed a little dry blood at the base a couple days later.  I did my research online, and found two common care methods: 

1) Do nothing, except keep dry and let heal naturally.

2) clean base with a Q-Tip and 70% rubbing alcohol.

I tried each method a couple days each.  The stump started falling off in pieces over the week, to which I was a little confused about.  It still seemed a little oozy, and I was not feeling confident in my care for it.  A little blood is supposed to be normal I found out, but it still didn't give me any comfort.

Finally, at my daughter's 2 week Doctor appointment my worry was cured!  Instead of the previous two methods, the pediatrician said to use Hydrogen Peroxide with a Q-Tip to clean around the stump, gently pat dry, and that's it!  Within a matter of a couple days, the remaining stump fell out, and my daughter's cute little belly button made its appearance!

DIAPER RASH

Our daughter has been a big healthy eater from the start.  In the first two weeks, she gained double the amount of weight most newborns gain.  However, her slightly larger appetite also means more dirty diapers.  After a couple days, I noticed her rectum getting a little red.  

Initially, I used Vaseline, but after a few more days the rash got worse and my daughter let me know it with each diaper change.  I needed to try something else.  I stopped using wipes, and switched to cotton balls and water to clean her skin.  I also gave her a warm bath to soothe her little bottom, and some diaper rash cream for the raw areas.

The trick that really started to help, was actually baby powder - even though I read many places online that advised against it.  My mom used it, and never had to deal with rashes for her kids.  So after cleansing my daughter's skin with water and cotton balls, I applied a little rash cream to the affected areas, then lightly put some powder in her fresh diaper.  After closing the diaper, I gave it a light pat to disperse the powder inside.  Within a day and a half, her rash and skin started looking better, as well as my daughter's disposition during diaper changes!

I understand that all babies are different, and have different skin sensitivities.  These are just two tips that have helped me.