If you could only play MP 3, then you should stay out of Torment entirely and play either Expert or Master, depending on your gear level. It shouldn't be too hard to upgrade enough gear to get up to Torment though; once you have over 100k DPS and 400k Toughness you should be able to do Torment I at least. The Toughness part is really the important part of that: my friend started today with about 70k DPS, and he could survive on Torment I, he just took a while to kill things. Monster health scaling is actually not that bad in regards to the shift between Torment I and VI; rather, enemies deal a lot more damage on the higher difficulties. To deal with higher difficulties, you should either have enough DPS and Toughness to survive, or have sub-par DPS but make up for it with high Healing (LoH, LoK, Life Regen, etc.).
You should be able to find some upgrades pretty quickly once you start, though. Of particular note is that the Archon armors will be very useful to you in terms of upgrading quickly and efficiently. All of them have been changed to inherently roll between 300-329 of the chosen stat, and all of them are smart drops by default (because all crafted gear is smart drops), so you can quickly boost your stats. I would say that you should be able to double your toughness at least with an expenditure of around 500k gold in crafting costs. If you are poor, and 500k gold is a lot, go finish quests. Each quest gives you about 30-40k gold, and crafting each piece costs 50-75k gold. Finally, to start crafting, you need Common Debris, which is the new crafting material from salvaging white items. White items are actually quite rare now, but they drop almost all the time from weapon/armor racks. As a result, the best place to get whites is Act I Cathedral and Act III Stonefort/Bridge of Korsik. Fortunately, each white/grey item salvages into 6-9 Common Debris, and each craft costs 15, so you should be able to rack up materials fairly quickly. During that time farming you'll also probably find some legendaries/sets, which at your level will automatically be upgrades even if they roll the worst possible rolls. Average rolls have been heightened significantly: whereas in the old D3 an item could roll 1-200 of a stat, in 2.0.1 it will now roll about 140-200. As a result, for your level, even the crappy rolls should still be serviceable, if not necessarily good.
Finally, in terms of what to upgrade first from crafting, I would say do Shoulders first, then Bracers, and then Gloves. Shoulders are cheap and cost very little crafting mats, and they are very easy to get good rolls from. Just get ~300 mainstat, ~200 Vit, some kind of AR/Armor and something else, and you're good to go. You should be able to get something like this in 3-5 crafts; less if you're lucky. Next is Bracers. The mainstat is guaranteed again, so all you really need is Vit and CC to be ready to go. However, Bracers can roll 'elemental skills of element X deal X% additional damage,' which is a very strong modifier if you are playing with many skills that share an element. As a result, if you can't find Bracers with CC but you can get one with decent additional elemental damage, you can probably stick with it. Finally, Gloves are hard to get a good upgrade on, since there are a lot of possible modifiers for them. Mainstat, Vit and CC are the best rolls, but since you're gear is not high-tier you should be able to find a good upgrade fairly quickly. If you are interested in playing a build with low cooldowns or resource costs, Gloves can roll cooldown reduction and resource cost reduction, so you might think about taking a roll with one of those but no CC. The Archon chestpiece can roll nicely, but it is the most expensive of them and costs a lot of materials, so you might not want to risk it.
This guide should get you at least to the point of being able to be comfortable around Master difficulty, which should be enough to get you started. One final piece of advice is to try to play in a party with people. You get more experience thanks to the multi-player buff, and since you can only trade with people who found a legendary in the same game with you, you'll have a better chance of trading for something that you can use.
