The major LucasArts games are worth starting with. A number of them have remasters and mechanically play as well as they did when released. Ironically it's the 'newer' 3D ones which age worse in terms of presentation and control, but most are still serviceable today, due to the nature of the gameplay.
If you're interested in looking at the classics, I'd say start with Day of the Tentacle, Sam and Max Hit the Road, Indiana Jones and the Fate of Atlantis and the first three Monkey Island games. If those grab you, it may well be worth checking out the rest of their back catalogue. Theres a lot of good stuff in there.
Sierra have some great classics too. I find them harder to recommend straight off because the SCI interface isnt quite a nice as LucasArts SCUMM engine, and they have a nasty habit of punishing the player by killing them or putting them into a dead-end/unwinnable state. This can be a frustration. If you're up for the challenge though, go for it. Gabriel Knight: Sins of the Fathers is one of their better games, and one of my favourites.
Other general classics have been mentioned already, but to echo: The Broken Sword series is excellent, and Beneath a Steel Sky is an earlier on from Revolution which is pretty good too. Toonstruck is great too, and Bladerunner, which has just been rereleased on gog.
Someone already mentioned UHS. I think that's a good shout as it gives you some degree to control over how much help you want. It's a good idea to have it for when you're stuck, as it can happen even in the better games and it is a frustration point when you just want to keep on with the story.
Have fun if you do start digging into this stuff.