For now coastally, you have relatively little to worry about.
VHF is fine. You aren't going 30 miles
offshore yet are you? And VHF radios now have an
emergency button to push that will send out a
distress signal along with your
GPS coordinates. Become a
member of BoatUS and it's like AAA of the sea. They will come and give you a tow. If a sailboat engine breaks down, you can still sail... or just relax and wait for the
wind or call for a tow. There are satphones but you can get an
EPIRB which is an
emergency position indicating
radio beacon which alerts planes and satellites worldwide which relay the message to Coast Guard for help in case of emergency; you just flip a switch. You can get a
liferaft, nowadays they are very tough and reliable. These things can be a start to calming the fears of your partner. Then just go out a mile or two and come back in. Once you see that you don't sink or have any problems you will venture farther out and the whole thing becomes a lot clearer. And then you realize that even in a leaky
dinghy you are safer on the
water than you were in the car on the highway that got you to the harbor in the first place.
As for types of emergencies... well I tend to bump my
head on things a lot, but that is just me. Slipping and falling. Sunburn.
Fish hooks in the finger. Knife cuts. Now if you are talking about rough
weather, then maybe things
ramp up a little, but you aren't going out in that yet. Weather forecasts are fairly good these days too.
And I believe there is a "Sailing for Dummies" book too. Try not to get permanently tied to the
dock, there is a lot of fun out there!