Did you start a seasonal character or non-season? On season, you get a free set (and lots of people to play with - so you might actually restart on season if you haven't, it's going to be worth it in the long run). Here's a starter build including a video - as you can see any item works, but it assumes that you got the free Earth set from the seasonal mail.
If you don't have that, I can't really give you any particular barb-specific advice as I've never played the class without sets, maybe someone else can. Just to add one thing: You should definitely play on Torment 1 even if it's a bit slow, because only on Torment you get the class sets and getting one of those is key to tackle higher difficulties. In terms of bounties, always make sure to run them on T1 (in the beginning) or T7 (later on, when you're really strong) - see this difficulty chart. The main reason you run bounties is for cache materials and they're the same on T1 to T6. And I'm just going to mention here, in case you don't know already, that you should always do the bonus bounty and complete the act such that the bonus tag switches to the next act, so that you get 2 caches from every act.
If you play non-season and decide to stay there, just try to get a set and some well-rolled legendaries as soon as possible to be able to venture into higher Torments. Run one bounty run on T1 when you feel you have 3 viable items to extract, but besides that focus on rifts and Greater Rifts (get those legendary gems and level them - usually Bane of the Powerful is your first gem and good to have on 25, after that Bane of the Trapped as high as possible, and a third gem of your choice; Boon of the Hoarder, Wreath of Lightning, Esoteric Alteration are some default options).
If you play seasons or decide to restart on seasons, you'll get two free set pieces from reaching level 70, another 2 from killing Kulle on Torment 2, and then the last two from completing a Greater Rift 20 solo (which is about Torment 4 difficulty, so if you can finish a T4 rift in about 10 minutes you should be good to go for that).
I decided not to play seasonal because I thought it was a good idea to have one non seasonal char to access adventure mode, and then play a seasonal one. But truth be told i dont 100% understand what season is (because it doesnt exist on console) so i have to read up on it.
Thanks for all the tips, i will make sure to put it to good use