I want to update my game to the 1.7.2 update, but I am wondering, does it cost money to download a new launcher?
No. Once you have purchased Minecraft once, you own all future versions of the game on that particular platform. Updating will never cost anything.
Worth noting is that Mojang have only ever explicitly said that future versions will never cost anything for users who purchased the alpha version. However, no update so far has ever costed anything for anyone, and Mojang have earlier voiced the opinion that updates never should be paid.
Minecraft's launcher has never cost money – it's getting a Minecraft premium account that costs. (If you have ever paid for just the launcher, I'm afraid you were probably scammed.) A premium account is required to login with the new launcher.
The new launcher can be downloaded from Minecraft.net. This is the exact same place the old launcher was downloaded from.