https://www.youtube.com/watch?v=xV-_DkpWu_s
I would check out some old westballz vs. leffen videos to get a really good idea of how to move around on battlefield. The gist is you want to shine -> wd towards where you hit them -> immediately shine them again -> wd down towards the closest platform. The higher you get (i.e. top platform), the easier it is to rack on the damage, so most of the time I will try to shine them two or three times before I use an aerial.
Falco's combo game is very dynamic, and a "flow chart" is not really applicable. There are just too many tech/DI scenarios. The best way to get better at comboing with Falco is to savagely grind Battlefield w bots and netplay. At the same time, there are some things you can do to make your life simpler.
Once you get your opponent to top platform height or above, you can do a doublejump fastfall dair and aim yourself to land around the inside corner of the side platform your opponent is flying towards. By placing yourself in the inmost corner, you can cover several options with minimal movement. Uptilt and shielddrop shine (something you should REALLY PRACTICE as both as a combo and defensive tool) are supreme in that they are combo extenders that allow you to stand in place. If they tech away, you can wavedash forward shield pressure, wdf shine grab, or wdf drop down shine if you slide past them (your body will push them off of the platform and your shine will catch them).
If you end up landing in the center of the platform, you can catch them out with a low percent F-smash if they tech in either direction. Since their rolls are longer than the length of the platform, you can do this on reaction. If they tech mistech, you can shielddrop shine and wd down to continue the combo.
These are just starting points for learning Falco's combo game, but they are SUPER important and cannot be neglected. Although the bird has the tools to win neutral 10 times in a row, you should try to minimize the needed openings for a kill.