I was noticing that my platy's stomach was getting bigger and I thought I was over feeding. Then today her stomach was huge so I looked up what a pregnant platy looks like and she is for sure pregnant! This is my first aquarium (10 gal) and I have no experience of what to do and how to manage this.
You have several options, depending on how much effort you want to put into it and what you want your end goal to be. If you decided you want to keep the babies, even for a while, you then have to decide if you want to keep them in the main tank, if you want to put them into a separate tank, and if you want to protect them all or let them fend for themselves.
Lets say that you want to have them in a separate tank. I think this is the best option as you can keep all the babies healthy until you decide if you want to keep them or give them away to an individual or a pet store. I use a 5gal hex I got cheaply at Wal-Mart with a piece of cheese cloth or panty hose rubber banded over the built in filters intake. Babies are weak for a couple of weeks and can be sucked against the filter intake and killed. The cloth reduces the flow and prevents this. You'll also need a light, which was built into this and a heater. Looks and effort are up to you, but I don't decorate a baby tank. I just let a bunch of plastic plants free float.
Assuming you don't have the time, money, space, inclination, or permission to have a separate tank, you can certainly keep them in the tank you have. If you do that, you have a couple of options. One is to make sure there are a bunch of plants for the babies to hide in. Besides the tall leafy plants, they sell little ones that have small spaces for the babies. As long as they have places to hide and dodge the big fish, they're very agile and the adults don't pursue them too hard, since they're being fed.
You can either let your mother just have her babies in the large tank and not do anything else, or you can capture them by using a livebearer birthing trap.
You wait till your fish looks like it's really close to giving birth, like when it's belly seen from directly behind goes from roundish to squarish. Then you net them, put them in here, and let them have the babies. The babies are too weak to swim and fall down the V-trap, through a slot at the bottom, and are caught underneath where neither the mother nor the other fish in the tank can eat them. You can then either transfer them to a separate tank or remove the mother and the V-traps, and keep them in there until they get stronger, larger, can eat adult food, etc...
I've had success with all methods. Sometimes you just don't get the timing right and the mother births in the main tank. It's surprising how many survive. Also, you don't have to deal with culling. You usually get some fish that are born deformed and it's much easier to have another fish take them out quickly than to have to do it yourself or to have them die slowly from whatever is wrong with them.
As for feeding, it's harder to feed them when they're in with the adults, since they have to expose themselves to get the flakes. Also, the flakes are too big and they have a hard time breaking them off. I typically spend a good amount of time crushing the adult flakes into powder between my fingers and storing it in an old beta food container. It'll last you a long time. I feed this to my babies till they can tear off chunks of the big food.
Lastly, another option is to add a tank divider to 1/4 of the tank.
This is one option and you can get them where the babies can swim back and forth through them or where they're completely segregated with only water exchange allowed. This gives you a little of both worlds, where you don't have to set up a separate tank, but you can separate them and give them a safe place to grow, also being able to feed them separately.
Platys are very easy, you don't have to help much. Babies are born living and can swim from the first minute. If there's something to hide, they will survive. If not, other platy will eat them.
If you're interested in having young platys, take care about what you do if the grow up. Your aquarium might be too small. Calculate one inch of fish length per gallon and take care to give them away as soon as they're 3/4 of the full size of a parent.
If you don't like babies, just remove everything to hide, the others will hunt'em and eat the babies.