How does one most effectively progress from Torment I upwards?

If you keep doing what you're doing right now, you're on the right path. Remember that it takes some time and luck to make a character strong.

To answer your question about quality and frequency of drops in difficulties:

• There are some torment-only legendaries that will only drop from torment 1 and above
• Each torment level increases the chance of getting a legendary. The quality of the legendaries will only be affected indirectly. Since more legendaries drop you will also get more ancient legendaries which are a lot stronger but also more "bad" legendaries.
• Crafted legendaries can proc as ancient ones but it's very rare and costs a lot of resources. Most legendaries have partly fixed and partly random stats. These rolls are not affected by difficulty. (The resource drops in order to craft them are though)
• Gambled items also have a chance to be legendary but the chance doesn't increase with difficulty. Stats and items are completely random although the type of the legendary is restricted.
• Some enemies like greater rift bosses and certain treasure goblins have an increased legendary drop chance.

Your main goal should be grinding for legendaries and/or sets (usually luck based) in order to boost your damage and defense but there are some tricks you can do gain a good boost in dmg/toughness that aren't too luck based. Personally I wouldn not recommend crafting unless you're getting really unlucky since it costs too many resources you could otherwise spend on reforging your gear.

Seasonal Journey

Now if you're playing a seasonal character you can complete your season journey in order to get a "free" set which helps a lot (note that you will get a set only once per season).

Most sets you get from the journey are built like this:

• 2pc bonus: Usually a mediocre damage upgrade that allows you to rush through torment 1-2
• 4pc bonus: A strong defensive boost that helps with surviving on torment 2-6. At this point torment 3 and/or 4 are your preferred difficulties since your damage isn't really good enough for 5-6.
• 6pc bonus: This is a huge damage boost in most cases. We're talking about 1000-5000% more dmg than before. With this damage you can start grinding Torment 6-7.

After this your main upgrades will come from finding (specific) ancient legendary Items and upgrading your legendary gems.

Gems

Gems are also a great Way to increase your damage. Having a high level green gem in your weapon will boost your damage by a ton. It's really easy to farm specific gems with the Broken Crown which is a fairly common item. Especially if you have access to the greed portal. Your main-stat gems in normal armor will give you a good boost but you should always prioritize getting the 130% more crit dmg on your weaponslot.

Non seasonal characters / alts

If you don't play on a seasonal character or you've already completed the seasonal journey all you can really do is grind, grind, grind. The things I've mentioned above should help you getting your legendaries faster but you won't be able to progress as fast as with a "free" set.

While not a full answer to your question, I feel this is worth adding:

One of the best places to farm legendary items is in Greed's Realm, as Jutschge mentioned. This also gives you plenty of gold, which you will need for enchanting to optimize your builds. This Realm can be easily accessed by breaking down Puzzle Rings in Kanai's cube, allowing you a one time entrance to the Realm per ring. Plenty of treasure goblins are inside, and if you can kill them before they escape, you've got a good chance of them dropping a legendary or two. The final boss spawns treasure fiends as well, so it's wise to kill as many as you desire before killing Greed to get even more loot. Greed gives a legendary gem, as well as a fantastic amount of loot, further increasing your chances for legendary items.

First I would recommend asking someone in general to run you through a bunch of higher level nephalim rifts to get you some paragon points and gear. It's late enough in the season that most people have a decent character, and some are willing to help. The legendary and set drop rates are increased the higher the torment, so you will be swimming in useless legendaries (and some good ones) after a couple of runs.

You don't always want to increase your torment level. At some point you are going to lose out too much on speed and it's not worth it to increase the torment. However, T1 is low, you should at least be aiming for T10.

In regards to actually getting there, you want to focus on a specific damage type and then buff it as much as possible. If you haven't finished the season journey to get your free set do that. It will provide you a great buff and should put you around T5 or 6 with just that set's buff.

For example, I got a Sunwuko set from my season journey its 6 piece buff adds 3000% damage bonus to Wave of Light, Lashing Tail Kick and Tempest Rush. I chose to go with wave of light so I also run Kyoshiro's blade, Pinto's pride, and Tzo Kirin's Gaze to all buff my Wave of Light damage.

What you want to do is look at the leaderboards (shift + L) to find what good builds are for your respective class. This will give you an idea of what you're looking for. Obviously, you won't have access to all the specific gear or the crazy high stat rolls, but if you go down the board you can find a lot of different builds. Once you find a build that you have the items for, look up that set on icy-veins to get a more in depth description of how to play it and what some of the alternative items are.