I think that the best way to farm for a new character is done best by following these steps:

a) Use your shards to gamble for a weapon. If you have an Ancient Legendary weapon that you can give to your weak character, that would be best. Use your materials to make some starter gear for the new class and start playing it. Unless you are playing a class that shares the same attributes, the smart loot system can give you stats that you don't need on the new character. Farm on the new character after you make some good stuff for him/her.

b) Always push the highest difficulty mode that you can do efficiently. If you are taking 20 seconds to kill a monster, that is too long. Aim for fast kills with a fluid run that gives the highest drop rate.

c) Some builds require items that only drop from bounty crates. Make sure you know what you are farming for.

d) Try to get an Ancient Legendary weapon first (even if it isn't the greatest for your build), because the higher damage will make up for the rest of your gear being garbage.

e) Complete your item sets first to get the good set bonuses before you focus on Ancient Legendary versions of the item. The set bonuses outweigh Ancient Legendary items in most cases (Legacy of Nightmares notwithstanding)

f) -- THIS ONE IS JUST MY OPINION -- GRs are for progression, paragon levels, and gem grinding. I find that you get more legendary items by doing T13 regular rifts. Do regular rifts for cash and gear...

I hope this helps.