None of this words can be used in every situation.
Ankündigung: Is an announcement of something that will happen, like announcing a speach, the arrival of someone, the arriving of someone to an event.
"Ankündigung" is mostly used for more important and/or official things.
Durchsage: is an announcement of something to a group of people, like announcing at a train station, that a train is coming in, or at a plane, that they are about to land.
Anzeige: An Anzeige can be better translated as a report than as an announcement, because it means to report something / someone to, for example, the police or other public authorities, but it isn't bound to the public, you can also use Anzeige as report when reporting your coworker to your boss. But it also can be a technical device to show information to a user, like Tankanzeige (fuel gauge) or Druckanzeige (pressure indicator).
A view more meanings of Anzeige by user1583209:
There are other meanings to "Anzeige" than the one you mention (="Strafanzeige", etc). The word "Anzeige" can also refer to "classified ads" in newspapers and other periodicals; It can also refer to displays (e.g. "Tankanzeige") or in medicine it has the same meaning as "Indikation" (indication).
Mitteilung: is a neutral translation for announcement, without much further indication, but can't be used interchangeably with Anzeige. Mitteilung also usually needs context if you want to use it properly.
Bekanntgabe: is again a more official announcement but a present for. while "Ankündigung" is an announcement of something that will happen, Bekanntgabe is about things that are happening, of course the context can change this again.
Ansage: is a more direkt and less formal announcement than Bekanntgabe.
While there is no short way to learn these things (at least I don't know any) you can cheat a little by taking these words apart, every Word here is made up out of other words, that give indications of what is meant. For Example: Durchsage contains durch und sage, through and saying, so it is about saying something that is supposed to get through to people.
But be careful many of these words are old and the origins may not be used any more or in another context.