When does it become necessary to start crafting/trading etc?

I just started playing less than 2 weeks ago during the middle of the month so I'm still relatively new. I'm level 44 right now (just now about to fight garuda story-wise) and basically only have thaumaturge leveled with 15 points in archer just for black mage. I've gotten through the game up to now only using gear from quests/dungeon drops, and I haven't even touched any of the disciple of the hand/land classes.

I'm asking because I noticed a pretty big difficulty spike recently, particularly in solo instanced battles that started about when I first went to Ishgard. There have been a few times where I got literally one-shot (steropes FATE, cutters cry chimera and some stone vigil bosses hurt a lot). I've just noticed other players around my level with blue/high quality items (an explanation on those colors would be helpful) and I'm wondering if I should have leveled other classes by now?

Only if you wanted to. Crafting/gathering is not necessary unless you are determined to make all of your own gear (blue gear is almost all token turn ins from high level/endgame stuff though).

I honestly didn't seriously level a second class till I was about... maybe on the 2.2 content stuff, so well past hitting 50 (this was before HW). I dabbled in a few other classes but nothing serious. It's not necessary to level any other classes unless you want to play them. For BLM I think there may be a few other cross class skills from ARC an ACN but I can't think of them since I don't play BLM...sorry.. but it's not required.

First, welcome to the game.

Second: You misunderstand how powerful crafting is. Crafted items at best, are Greens, and HQ versions of items are only about 2% better than the same normal item. Crafting/Gathering is useful if you want to make some extra money on the side or are interested in something besides raiding, but there's no huge benefit from it.

The general order of power for items, in order of least powerful to most: Grey < White < HQ White < Pink (Dungeon drops) < Green (Dungeon rares) < Blue (Token items/Raid items) < Purple (Uniques/Relic Quests)

Crafted items at best are Greens. Blues and Purples are much more powerful, and can only be obtained by trading in endgame tokens from raids or "expert dungeons".

If someone in your group is running with Blues, they are using gear that is effectively level 120 or 200. Compared to your level 44 gear, it's obvious they'll have a lead.

As for the difficulty spike, that is to be expected. Black Mages are one of the most fragile classes in the game. If you are having a lot of trouble surviving, I would double check to make sure your weapons and armor are within 5 levels of your current level. If they're not, you need to look in to replacing that equipment right away. You can craft your own items or buy from the market board. If it's too expensive there, your grand company will have items you can dump company seals in to if you need a green item to fill a slot until you get that dungeon drop.

Black Mage also has a lot of barrier tools (Magic negation via manaward at lv.30 and Physical negation at lv.45 via mana wall) that you should be using to prevent damage. Those tools are so powerful they let you take a tankbuster (attack designed to kill a tank) without a scratch in raids. It is a 120 second cooldown though, so you can't exactly spam the things, but it lets you survive in case you mess up or have to complete a cast for your rotation.

Also, if you are planning to play a blackmage, I would highly recommend getting archer to lv.34 for Quelling Strikes before going past level 50. You're still leveling and undergeared for now, but once you start collecting tokens and become equally geared at level 50, you will notice how quickly your attacks pull enmity. The ability to reduce your enmity will come in handy then. Otherwise, level Arcanist to 34 before you hit 60 and run savage raids (hard mode raid dungeons).

Hope that helps! Be sure to keep us updated if you can.

You'll get your first blue-quality gear from your level 45 job quest, so you'll see a significant boost in your stats at that point. Then another and much bigger stat boost will come when you hit 50, which unlocks gear with item levels ranging from 50 to 130. (If you'd been in the game since the beginning, you'd have gone through quite a few level ranges in between there, using lower iLevel gear while you worked on getting the next tier, but extra gear rewards have been added to help newer players catch up, so you can now skip over most of the levels in between unless you want to track them down for cosmetics. Once you hit 50, you should now be able to jump right to iLvl 90 gear while you collect tokens for iLvl 120 or buy iLvl 115 stuff.)

Crafting is never necessary, but is quite helpful. Bosses like steropes and chimera are rough because they have non-telegraphed aoes that hit hard - learning the patterns and dodging are most important there.

However, the biggest advantage of having crafters leveled is you can repair your own gear - it's cheaper and can be done anywhere, including in combat. As long as you have dark matter, forgetting to repair and having all your gear break in a duty is not a problem.

The biggest advantage to crafting is gil.
It is a very addictive game within FFXIV.

Here's a guide on how to get started.
https://www.youtube.com/watch?v=IcNeKh8s0xs

There are always lots of deaths in the Steropes fate because it has that giant aoe that isn't telegraphed and oneshots pretty much everyone except tanks. There's another fate in Northern Thanalan that has a chimera that is much the same. Lots of fun to be a healer in those. XD

There are a few difficult story instances, especially once you get to coerthas. Remember to have some potions, level up Arcanist a bit so you can cross class physick and heal yourself in a pinch. Don't worry about dying to steropes or any of the big boss fates, they need to be done with a pretty big group, they aren't really solo-able like some of the fates with little mobs. And yea, being a black mage you are as squishy as they come, so a lot of things will one shot you that other dps will survive.

Crafting and gathering is not necessary but does come in very handy and will potentially earn/save you a lot of gil over time.