I would start with the original, of course. Other good entry points are OoT and A Link to the Past. I recently picked up WWHD and I couldn't help but think it would be a good starter game as well.

If you are going for the top down games, start with A Link to the Past. It is an amazing game and it should be played before A Link Between Worlds.

If you are going for the 3D games, you can really start anywhere. They are all great jumping in points and they are all great games.

I say go back and play the older handheld titles after you play a few other Zelda games

EDIT: I will add that the Wii and Wii U are the only places you can play every console zelda, but Wii U has Wind Waker HD (as opposed to the original GC game) and will be able to play the upcoming Zelda games....
I'm not 100% sure about Europe but you should be able to play all the games either on the Wii U VC or the Wii VC (which the Wii U can use)

I love Zelda, but I've never actually beaten a Zelda game GASP! I probably love skyward sword the most. I just couldn't get anywhere in Twilight Princess for some reason, even though I did like that game a lot. I do plan to someday beat these games...

Well, then it's settled. I'll buy a WiiU. Hopefully, the releases of Mario Kart 8, Bayonetta 2, X and SSB will resurrect it, because it's still a great console.

If you want to play Majora's Mask, make sure you play Ocarina of Time first. The story will make more sense that way. Other than that the stories are self-contained. I'd also play A Link to the Past before A Link Between Worlds if you're planning to get a 3DS or already have one, it's really cool to experience how that world changed in some hundred years.

But yeah, other than that, start with the game you feel you'd like to play the most right now.

And yeah, get a Wii U. 

Play Ocarina of Time 3D. Thats about as good a place as any to start the series. If not that, then A Link To the Past should be on the 3DS eshop by now. It'd be a much smaller investment, but the game is still a masterpiece.

This man speaks the truth.

I haven't played a Zelda game for more than 20 minutes by myself. The furthest I got was the water temple in OOT but that was mainly watching my friend do it for me while I watched because I didn't understand where to go...

Zelda & Metroid are so highly rated by the Nintendo fan base but I don't care for either of them.

The Wind Waker did come bundled with my Wii U but I haven't touched it yet...should I actually try to play it?

For me, replayability comes to how much do I want to erase my save file and play over the story again because of how awesome the experience was. For twilight Princess, I do this every summer when I don't have to worry about school. I will admit Hyrule field is pretty empty. But the game at least has a lot of collectibles like Poe souls and bugs and way more heart containers. You can also try to find find all the hidden skills but you can find all of them before beating the final boss.

When did you first play Ocarina of Time? At the 90s and before other 3D Zeldas? Because there is no doubt Ocarina of Time's polygonal graphics and slow frame rate has aged horribly for today's standard. The dungeon and the bosses are great. But if you can't get past the game's graphic and slow sword game play, you'll be hard pressed to finish the game unless you're a big Zelda fan.

Do you feel proud of that atrocity? Not playing OoT? TP?