I'm currently studying Kanji using a flashcard app on my computer. I do know that there are two readings for each character, On and Kun. What I don't understand, is when a Kanji has more than two readings. Three or four readings I can remember, but once I get to more readings with say 10 readings, that both confuses and concerns me. Say for the Kanji 上. I know its three On readings: ジョウ ショウ, and シャン. but the Kun readings scare me. うえ、 -うえ、 うわ-、 かみ、 あ.げる、 -あ.げる、 あ.がる、 -あ.がる、 あ.がり、 -あ.がり、 のぼ.る、 のぼ.り、 のぼ.せる、 のぼ.す、 たてまつ.る. My question is when do I use these readings, and do I need to know all of them?
The linked post from ConMan in the comments (Do we really need to remember the kunyomi and onyomi reading of each kanji?) goes most of the way to answering your questions. But, to directly answer them:
When do you use these readings? -- when the specific piece of vocabulary that is being used demands it. That is to say, you can't know for sure unless you know the piece of vocabulary in its context.
You might be able to guess the reading if you have seen the character used in another word (this applies more frequently to the on-yomi, as "kanji compound" words are numerous and typically use the on-yomi of the given character, e.g. knowing 両親（りょうしん） might enable me to guess the reading of 親 in 親友（しんゆう)), or perhaps if you know the reading of a certain component within the character (e.g. the 師 in 教師（きょうし) shares a reading with its component 市). But this approach is not foolproof, so it is not possible to abstract a general rule which dictates when to use a given reading over another. It simply comes down to the piece of vocabulary in context.
You might initially think this is absolutely crazy, but the same can be said of English: knowing when 'lead' is read 'lead' and not 'lead', or 'read' is read 'read' and not 'read' all comes down to context.
Do you need to know all of them? -- this obviously depends on your personal goals and needs, but assuming these are within the realm of normality... the short answer is 'no'; the long answer is 'It depends on the character, and the specific reading you are talking about. Some of the readings for characters with many many readings will likely be used more rarely, and so you may well find your Japanese learning journey proceeds fine without them. But, it might be unwise to not know all of the readings for other characters where all readings occur in commonly used vocabulary.'
This answer perhaps obscures the main point, however, that how commonly used a reading is depends on how commonly used the vocabulary (whether a singular word or multiple words) is that uses the reading -- so you might be better to reframe your thinking as: 'do I need to know this piece of vocabulary, or not?' -- in doing so, you'll (hopefully) naturally build up a knowledge of the readings of given characters that actually matter to you.
For what it's worth, I have never sat down to specifically learn readings of individual characters myself. I try to learn the vocabulary that will help me most, in a given sentence/context, and then hope that those pieces of vocabulary stick, and that common readings of characters emerge. I'm not saying that sitting down to try and learn a character's readings is totally unproductive, but I personally think (and I think this is a widely held view) that it's more effective to learn actual words that character readings in isolation.