Strawberries are the easiest for me. I have June bearing strawberries, which gointo heavy fruiting one a year. Great for jam making, ice cream, shortcake.
I also have some ever bearing strawberries in a strawberry jar and in some hanging baskets. Ornamental mostly, but it is fun to walk by and pick a juicey red berry to eat. Kids like this if you happen to be a grandparent.
At one time, I grew small Alpine berries as a border/ groundcover along the sidewalk. Tiny, sweet as sugar, about the size of your little fingernail.
All of these berries reproduce themselves, so once you buy them, you do not have to buy more, ever.
I also grow raspberries, in the second year since planting, the fruit is scarce, but I expect it to increase every year.
I am a country girl used to picking blackberries in the wild. But home grown blackberries are a treat. They are bred to be thornless and twice the size or native berries.
I live in zone 6, Southeat Missouri, USA. My neighbor grows a mango in his greenhouse. If you live in New Orleans, you may be able to grow a banana on the patio.
Citus fruit is good for Florida growers. Pineapple grows well in Hawaii, but it is strictly ornamental in my neck of the woods.
Apples and peaches are grown commercially near my home. I let the professionals take care of these tree fruits. I have grown a few apple in a home orchard. The spray schedule for peaches, pears, nectarines, apples is heavy and costly in the Midwest. However that is not a real concern in Wahington and Oregon.
The University Extenion in your state will have fruit growing guidelines. They are tried and true through the research of the university. Follow their suggestions to get started.