when I stayed in Mission Valley (we had a hotel the first few days when we moved here) they had hotel shuttles to SW and the zoo. You are about 40 minutes from the animal kingdom park.
In SD nothing is accessible. You need to rent a car or take some local tours.
http://www.sandiegotours.us/
You are an idea where there is nothing around BUT hotels. We don't have the public transportation they have in England.
Personally, I don't think any area should be avoided. If the area isn't safe then there is really no reason to go there. (i.e. tourist spots, local spots, trendy places to eat all great and safe). Lots of people will tell you this part of town or that but really they are naming places the average joe wouldn't be going to anyway.
You are going to have a blast. SD isn't really easy to get around but everyone is friendly and their are lots to do. Get a car or a couple tours and you will be fine.
La Jolla, the beach, (children's beach with the seals) and the torrey pines state park beach, There are lots of great places to eat. My favorite is Sbicca's in Del Mar if you want somewhere nice. But there are tons of really great places.
Some people will tell you not to go to Mexico. I might take that advice. The drug violence is pretty bad. The government broke up the big drug rings now the little guys are fighting for control. My husband who is a scientist has co-workers from Mexico who go back all the time. They speak the language and don't "stick out". So they think it is a bit funny that we thing it is dangerous. I think my husband saw "Man on Fire" one too many times.
You will need a way to get around. Have fun.
Ask follow up questions if you need to otherwise Good Luck.