I understand that an address is needed to send Bitcoins to other people, but I do not understand why you are recommended to generate a new address for each transaction. Also, I have read that an address can be saved for easy payment in the future, but I don't understand how this function would be possible if new addresses are generated every payment. Thanks so much for your time and help! It is much appreciated.
The simple answer is you have a keypair for each bitcoin address, public key & private key, you can think of this as a username (public key) and password (private key). You can share your username/public key with as many people as you'd like, whether you know/trust them or not, of course you must keep your private key to yourself. The interesting part is that the public key is derived from the private key, and you sign messages with your private key to prove that you know it without giving away what the private key actually is (this is verified by comparing your public key with the message you signed).
The technical way that you create a private key can be found at the offical bitcoin wiki here: https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses
This wiki has a LOT of information, and is the most accurate source on bitcoin related information I have found to date.
What they're saying is that you can use an address repeatedly, but you shouldn't because it hurts your privacy, and it hurts the privacy of anyone who sends a coins to a reused address.
New addresses are generated every payment as Change Addresses. As you may know, partial coins cannot be sent, just like you can't tear a banknote in half and give someone half of it. So every time you send someone money, unless it is a whole coin, your client will generate a new change address to receive change for. That is for sending money.
However, if you want to receive coins, you can just have a QR code that displays a single address that everyone can send to.
When I say "whole coins" and "partial coins", I don't mean ฿1, instead I mean the individual transactions' inputs and outputs. I could get into that if you ask.