OK.
A lot of games being recommended here ARE good, but they don't serve as very good introductions to the genre, especially to a modern gamer. Allow me to offer my own thoughts on the subject.
Most of the classic games from Telltale are a GREAT way to introduce yourself to the point-and-click adventure genre. The first two Sam & Max seasons (now known as Save the World and Through Time and Space) will serve you extremely well. They're a little pricier than your goal of $10, but they often go on sale, and the fourth episode is completely free [http://store.steampowered.com/app/8230/?snr=1_7_15__13], so please do go and take a look at that one.
If you've tried those and want more comedy adventure games, the already recommended Deponia series are fairly good (though a tad pricey when not on sale) and I will NEVER stop recommending Jack Keane, which frequently goes on sale for around $2. If you see it that cheap, for the love of Zod, GET IT.
If you're after something more serious though, create an account at GOG.com to get a couple of free games, one of which is the excellent future-set Beneath a Steel Sky. be warned - there are deaths in this, but they don't usually come from left-field.
Other serious adventure games include the Broken Sword series. There's been two traditional 2D games (both of which have gotten Director's Cuts/Remastered versions), a surprisingly good fan-game [http://www.brokensword25.com/] which fits after them, then 3D game that felt more like a lame platformer and another that used a mouse but wasn't particularly good. There's a fifth one that was Kickstartered and is coming out before the end of the year, so keep an eye out for that one as well.
There's a reason I haven't suggested any classic LucasArts/Sierra games yet, and that's because they're really not very good introductions to the genre. If, however, you want to know what all the old-schoolers among us idolise, you can't go much wrong with the recent remakes of Monkey Island 1 + 2. You can switch between modern and classic graphics and everything!
I would avoid Sierra games until you're REALLY comfortable with the genre. Not because they're bad, but because they have a different design philosophy than LucasArts games. LucasArts adventure games rarely feature the player dying. Sierra games will quite happily kill you for the simplest of things, which is why I've held off on them until now. They're good - don't get me wrong - but you need to be REALLY careful and save your game all the time when playing them.
And after that... well, the sky's the limit. There's no shortage of games out there if you're willing to look around a little.
Have fun!