I've never done so before in Skyrim, or most RPG games for that matter, but should I be investing into at least one of the crafting skills: Smithing, Alchemy or Enchanting?

Invest in all three and gradually you'll see how it dramatically affects your gameplay.
EDIT: Remember that the three skills will affect each other so it's best to level them all.

Does Legendary mean I can now obtain more perks? Just, I'm level 30 at the moment, so do you think it is too late to add it to my current character?

Investing in crafting skills will break the game.

No, it doesn't. Using all three does. Best to invest in one or two.

I did hear that most of them made you quite overpowered, which is why I steered away in the past. But, to be honest, I don't even play Skyrim for a challenge, which is why I usually play on moderate or lower difficulty settings. I play it just to immerse myself in the great world with a character with limitless opportunities.

Which would you say is the best? My character is a hybrid between one-handed combat and stealth using illusion magic.

Sounds like you'll want alchemy at least.

And investing into all crafting skills will break the game? How so? As in lagging/glitching or just too overpowered?

investing in all 3 doesnt break the game if you dont exploit it.

Poisons don't really appeal to me and I find so many potions as is, plus I do use a spot of restoration magic for healing, after all. I was thinking of smithing to get the good armour, or enchanting to get stealth related enchantments and boons to magic.

Not just poisons, you'll want healing potions, buff potions and fortify potions, especially fortify illusion when your spells start to fail.
Also, as long as you don't overly-exploit all three skills, it won't be that bad.
Not to mention Fortify Smithing and Enchanting will help with the other crafting skills.

They certainly do help, but by focusing on all 3 you can easily become overpowered.

Now if you were doing a Master Craftsmen build where you're effectiveness in combat was based soley on your equipment (meaning you rarely fight and NEVER invest in combat perks, offense or defense) then you won't be so overpowered. I did this with a Redguard and didn't start investing into combat perks until very very high level, and the moment I did the game became much easier even on Master difficulty (Legendary was not available yet). So say you become a Master Craftsmen, you would also invest in say Speech and basic Restoration, possibly Alteration (only using non-combative spells).

Will I be able to introduce them at level 30 though? This character is becoming a bit of a "jack-of-all-trades", which I hate.

Odds are, you're exploiting all three if you are using them.

Enchanting looks cool, but whenever I deconstruct an item to learn its enchantment - the effectiveness of said enchantment drops. For example, if I deconstruct something with 100% faster magic regeneration, then the enchantment I receive is only 10% faster. What perk is relevant to this?

Once a skill reaches 100 you can.

How does this whole "Legendary" skill thing work? It seems to me that you can get almost every perk in the game with it?

You just said it, Enchanting.

Yes, but what perks within it?

Everything in the middle.

Fairly basic stuff, I mean, even the first perk makes your enchantments stronger.

Disenchanting any item will reduce it to 10. Like you have a steel sword of the blaze (25 damage) , you disenchant it and you learn the "fire" enchantment with 10 damage. The size of your soul gems (Petty, Grand, Common etc.) determine how strong it is. Also, your perks you have REALLY MATTER. Take fire, frost and shock enchanter perks for damage, and insightfull and corpus enchanter perks for armor.

What Netch-Fletcher said. If you just hover over an enchantment, it will show a "base" magnitude. If you actually select it, it will change magnitude to reflect your Enchanting skill, perks, and the best soul gem in your inventory.

The relevant perks for magnitude are Enchanter (5/5), Insightful Enchanter, and Corpus Enchanter. Plus the left-side ones for elemental enchants.

Without Fortify Enchanting potions, the max Regenerate Magicka magnitude you can get is 50%. If you're talking about enchants like "Fortify Destruction and Regen Magicka," there's a "bug" (might be intentional) where the regen is fixed at 10%, no matter what your skill, but your skill increases the fortify part.

Investing heavily in crafting skills can make you powerful enough to trivialize every fight, except on Legendary. Here's a basic summary of the most important things the crafting skills can do:

Smithing: Increase damage and armor rating of gear

Enchanting: Increase damage and armor rating of gear by a multiplicative factor. Decrease cost of spells. Increase Alchemy and Smithing skill.

Alchemy: Increase damage and magnitude of spells. Increase Smithing and Enchanting skill (temporarily).

Just from that, it should be obvious that there are opportunities to interact the three skills. If you raise all of them to a high level and make full use of the interaction, you very easily hit the armor cap and can one-shot almost everything in the game on Master difficulty.

If you're mostly doing physical damage (and using Illusion on yourself, rather than enemies), it probably makes the most sense to invest in Smithing and Enchanting. Although at least *some* Alchemy would still be good for maximizing your Smithing power. Alchemy and Enchanting are pretty perk-hungry, though.

I'm currently limiting my PC to alchemy. Enchanting and smithing can get out of hand pretty quickly and easily. The drawback to alchemy is tracking ingredients and recipes because doing so takes time and requires a bit of organization. I personally enjoy it but I can see how it's not for everyone. Even limited to alchemy, a PC can quickly grind to 100 (tons of ingredients are available in stores, the College of Winterhold, various private caches such as Anise's cabin or Drelas' cottage). Hearthfire homes also provide optional gardens and greenhouses so you can stockpile ingredients like crazy.

I'm finding balance by strictly limiting potions that make sense for the character and only brewing enough for personal consumption, not grinding and stockpiling. So far, it's been really fun though I did indulge and get a Hearthfire house with full alchemical benefits (garden, greenhouse, fishery).

Maybe you can take this approach and just start with one craft instead of all simultaneously.

If you want the most powerful gear possible you'll want these perks -

http://chrizel.github.io/skyrim/#t/0/4w,0,m73im8,e0o,0,0,0,0

Of course, personal taste will play into smithing material choice if you like the look of certain armors. I spend a few extra perks in the light armor tree to wear the dawnguard armor with a leather helm because I like the look of a simplistic functional looking armor more than the elaborate/fantastical looking armors. Many people like ebony, which is a great looking set that's artistic and decorated looking but still sleek rather than gaudy/overdone.

And of course, some players want to limit their character's power. I don't see the point personally but I've never been a fan of Skyrim's combat style and never felt like it was a real challenge, just a matter of stats and abusing the limited AI.

There's some truth in this, but at the same time, the combat in Skyrim is broken no matter your approach. It all depends on how tedious you want it to be, or what difficulty you want to play on to convince yourself you're above the noobs on adept when really you're just spending more time doing the same thing with more cheese. You can beat any difficulty without any remarkable player skill and without ever dying if you know the game mechanics well enough to exploit them. Crafting just allows you to do it in a more casual manner, and faster, with less risk of a small mistake or bad luck getting you killed.