Definitely go to a Japanese restaurant that has a sushi bar. Don't try eating the fish you catch yourself, unless you know what your are doing. Generally, fish served in sushi restaurants has been deep frozen to help destroy any potential parasites. The temps of the fish get lower than any home or even standard commercial freezer. The fish is also kept at extremely low temperatures in a very controlled environment. Handling is done carefully as to maintain the fish as fresh as possible. By contrast, the best stuff in your supermarket wouldn't even be looked at by a reputable sushi restaurant.
When you find a place, sit down at the sushi bar and talk to the head sushi chef. Ask him if he buys the fish himself. If he does, that's good - means he cares enough to be picky about buying fish as opposed to just accepting whatever comes into the restaurant. All the best sushi chefs do their own buying by either going to the commercial fish market OR having established such a good relationship with the supplier, that the supplier knows the quality and type of fish the chef will buy.
Get into a conversation and you'll learn a lot. Don't be afraid to tell him that you're new to sushi and wanted to try it. In fact, them his you're new to sushi and you would like to order "omakase" That means "chef's choice" in Japanese. Since he'll know your new, the chef will more than likely be happy to start you out tame and work up from there. There are cooked varieties of sushi, including a roasted eel filet (very mild white fish with a teriyaki sauce). There is also the popular "California roll" which is cooked crab meat (or the fake stuff), avocado and rice rolled with seaweed. Another good one is "spider roll" which is a deep fried softshell crab rolled with thin strips of cucumber in seasoned rice and seaweed. From there, you can go on to mild, but flavorful fishes like tuna, salmon, or flounder. Don't be afraid to ask questions as GOOD sushi chefs love to talk about their craft. Kind of like fishermen loving to talk about fishing!
Good luck and have fun! I've been eating the stuff since I was knee-high to a grasshopper and its still some of my favorite type of food. Oh, one last thing, because of the extremely high quality of the fish, sushi is NOT cheap. The quality is key and you get what you pay for. For that reason, I would never eat at an "all-you-can-eat" place or a "fast food" sushi joint. Quality fish does not lend itself to such practices well.
JC