it is certainly a lot easier to get around with a car. Florida is hot and sticky so even if just for comfort, a car is desirable.
Delray Beach is a long narrow city stretching from the Ocean to the far West reaches in the county. The answer would depend on where you land. If you want to live East just off Atlantic Avenue there is plenty within walking distance - but a grocery store would be a little more difficult.
If you live west, you'll be closer to amenities like groceries, Target stores, etc but the distance between homes and such amenities will likely require transportation.
Buses are ok but the public transportation infrastructure in Palm Beach County isn't like NY/DC/Chicago
Some over 55 communities have shuttles to help people who don't drive that live there.
You could also consider looking to a more urban setting... CityPlace in West Palm Beach has all the amenities and you would hardly need a car... Boynton Beach has a new development where everything could be a walk called Renaissance Commons...
Boca Raton would be tougher but possible.
You can also look at more routes on this page. Below is another link to Downtowm Special Service. I hope these links help you. If you have more questions please call or email me.