Why would anyone buy an old treasury while the same treasury (for example 5 year note, two years note) is regularly auctioned during the year? I mean if the yield rise why to buy an older note instead of waiting some weeks for the next auction (I'm a beginner)
Because the price will change to so that the yield is "fair" (according to market expectations) and people want to use their money for other things now rather than waiting for the next auction.
On the day of a 2 year note auction, for example, if you hold a 5 year note that was issued 3 years ago (so it has 2 remaining years), the value of your 5 year note will adjust so that the yield to maturity matches the yield from the 2 year note auction. The price of the bond is constantly changing based on market expectations about future rates so the price of existing bonds remains fair (in accordance with the market consensus on future rates). Since prices remain fair, bond holders are free to buy and sell whenever rather than waiting and doing all their moves on an auction day. If you're saving money for a down payment on a house, you'd really rather not wait a few weeks for the various bond auctions to take place. And since prices remain fair, you don't have to.