Hi there (this is going to be long)
I lived in Yokosuka from 2003 to 2006. I absolutely loved it and had a great time. Japan itself has a lot of beautiful places to visit and MWR does a great job of offering a variety of events and trips that are more budget friendly than doing it yourself. If you are more of an adventurer, the train systems are quite easy to navigate once you get used to it.
Traffic in Yokosuka and between other cities (ie Tokyo, Yokohama) can be a little trying. I lived about 6 miles off base and the commute was anywhere from 40-60 mins in the morning (yes, I'm serious). Just a lot of people going in the same direction

The housing office will help you in finding a place to rent out in town because the housing waitlist tends to be quite long. We lived out in town for about a year and a half but the washing machine that the base loans out was just too small for my husband's clothes (sounds crazy but it drove me nuts). Forget about using the dryer, you have to hang your clothes and in the winter, your house can look like a laundromat. I would advise getting on the base list as soon as you get there but enjoy your experience living out in town while you can.
The Yokosuka area was VERY safe when we were there, unfortunately at the very end of our tour, there was a rash of crime but it was mainly the sailors from the ships. The base CO instituted a curfew and cracked down, I think it got better, I never heard anything further about it from my friends that were still there.
Shopping out in town can be very expensive, I basically shopped at the exchange or online for clothes and Christmas presents. Also, there are some great little food markets but for your day to day groceries, go to the commissary.
All in all, I loved Japan and if my husband wasn't retiring I would take another tour there. Make sure you ask for directions to the "D2". Its a store similar to Wal-Mart and has lots of stuff for great prices.