Here's how I would do it:
Try to find apple juice with no preservatives, from a local orchard if you can.
I like to pasteurize the juice at about 150 F for 20 minutes, then let cool, then add the yeast.
Try Windsor ale yeast or a Scottish ale yeast for low attenuation / high final gravity and sweetness, which will help balance the strong acidity of the fruit. For 1 gallon, you'll only need like 1/8 of a packet. Save the rest in your refrigerator for next time.
Ferment just the apple juice by itself for 1 week, then rack to secondary.
Then ferment for about another 2 or 3 weeks or until it stops fizzing. Keep it as close to 50-55 F as you can the whole time. If room temperature is the best you can do, that's alright too. But cooler is better in my experience, if you can.
When it seems to be done fizzing, then get your raspberries ready. You'll need about one pound, fresh or frozen. Crush one Campden tablet and add to the berries, and then leave them sit refrigerated for at least 24 hours if not 48 hours. Then rack the cider on top of the berries in another fermenter, then leave it for another week or two until it quits fizzing again.
Finally, on the day you bottle, add about 1/4 cup of this:
This syrup will add your mango flavor AND act as your priming sugar, without the fuss and mess of racking onto real mango chunks, which would be fine too... Alternatively, you could go ahead and use real mango at the same time with the berries, and finally prime with 2 tablespoons of cane sugar instead on bottling day. Either way.
I think that should do the trick. Good luck and enjoy!
