I have successfully raised one baby betta, I have 2 who I would say are "juveniles", old enough to be sexed but still small and developing colors/fins, and three babies. I tend to do very well with them.
I keep their tanks at about 80 degrees, sometimes a degree or two warmer than that (I don't have a heater in the tanks currently, I'm in HOT Iowa and our A/C is only so-so), but I do monitor the temps several times per day. For food I've found that the Hikari (I think that's the brand) small bites work the best for the little guys. Once their mouths are big enough I give them the Aqueon betta pellets all my other bettas get (I will just randomly try a pellet in the tanks when it appears their mouths might be big enough).
I have found that they tend to get stressed when given too big of a tank when they're so little. I keep my little guys in a half gallon unfiltered tank. Once they get a little bigger and a little stronger I will move them to a bigger tank with filter. In the meantime, I just change water every day. It's easy to do in a half gallon and I do try to minimize stress by using my hand to "cup" them and they put them in a little cup to change the water. When changed I just place the little cup on it's side in the tank and let them swim out on their own.
They really are a lot of fun to watch them grow, change colors (I think all of mine have turned out to be completely different than what I originally thought they would be), and grow fins. I love to guess what kind of bettas they will be (I'm usually proved wrong). :) Good luck and have fun! Don't hesitate to ask if you have any other questions.