As /u/bradamantium92 said, you can ignore the in-series timeline (what you posted is the release order) because it's very complicated. Once you get into it, you'll definitely want to look into it though. There are lots of great Youtube videos explaining the world and its canon.
I'd say A Link to the Past (SNES/Wii U via VC) is an enormous must-play and probably a great place to start. This is one of my favorites in the entire series. The dungeons are inspired and well-designed, the story is well done, and the progression really works. Plus, it holds up REMARKABLY well (as many well-designed SNES games do, imho) and immediately presents what I consider to be the real heart of the series: exploration, dungeons, item and power-up collection, puzzles, and strategic combat. If you really like it, I'd also suggest A Link Between Worlds (3DS) because it's a wonderfully done spiritual successor. The former is a classic, the latter is a great re-imagining.
If you don't really like LttP, perhaps the 3D games are more your style. Ocarina of Time (N64, 3DS remake, Wii via VC) is, of course, widely known as a great game. Just with Mario 64 if you're playing it in 2015 for the first time, you have to remember that this was the outset of 3D gaming. Most people find it really easy to pardon the graphics, but you've also got to think about how new the movement and camera controls are, and things like that. This isn't actually anywhere near my favorite game personally, but a lot of people love it. It's also great to play right after LttP because it was the very next console Zelda game. If you like it a lot, or perhaps even if you don't, also go for Majora's Mask (N64, 3DS remake, Wii via VC). It's easy to tell it was made with the same assets more or less, but the story is a wild departure from the normal Zelda fare. This game comes chronologically after OoT (with complications that you'll understand once you play if you're not already familiar).
I'll add one last strong suggestion, and that's Wind Waker (GC, Wii U remake). It's made a much better game after playing OoT because this is also a chronological "sequel" in some ways to that game (though after a really long amount of time, and again with complications that will make more sense after OoT). The art style is yet again very different. Some people don't like the cutesy/silly style, but I personally love it. The game itself is also really well designed. The story-telling won't floor you in many places this time around, but there are still some really cool bits. Exploration, though, is really rewarding (though you'll also grow tired of just getting rupees in a lot of sections -- I feel this game would have benefited from some added collectibles for the explorer-types).
I notably left out Legend of Zelda (NES, 3DS via VC), but that's just for ordering reasons. Above I've given you a more-or-less ordered playlist, with optional titles you can skip if you didn't like what preceded it. I don't think LoZ gives you a great introduction to the series in 2015 because it's just so very hard (for me, at least) to put my head in the mindset of games from that era. At the time the story was revolutionary, but in comparison it's one of the sparsest stories in the series. The game is punishingly difficult as most games of the time were because they didn't have a lot of memory but they wanted the game to have longevity. However, it's still definitely a great game, it's just hard to say when, precisely, to play it. LttP will probably give you the best sense of if you'll like the original LoZ because their spirits are largely the same: exploration is key, and experimentation is rewarded. LttP cuts a lot of grinding for secrets by giving the player clues that LoZ lacked. You may want to do what I did and play LoZ normally, but with a walkthrough for secrets -- trying to burn every single tree to find a single dude who gives you 50 rupees just isn't fun when you want to play the whole series in a summer. Just definitely don't use guides for the main part of the game, especially the wonderfully designed dungeons.
I could keep going on and on about Zelda, but this is definitely too long already. LoZ, LttP, OoT, and WW give you a sense of four of the biggest games in the series. I didn't touch on the late-GC-to-present era (Twilight Princess and Skyward Sword being the two biggest entries, with hella hype for the upcoming Wii U game) mostly because I haven't played it extensively, but there are some different styles there that some people like and some people hate. Those four games though, to me, create the heart of the series. Other games (Zelda II on NES, MM and LBW which I mentioned, the DS games Phantom Hourglass and Spirit Tracks in the style of WW, and so on) are also good, but to me are better left to once you're already hooked on the series. And once you're familiar with the characters, stories, and themes, even spin-off games that are of completely different genres will draw you in, like Hyrule Warriors did me.