Another way to quickly get some of the tier pieces is Blood Shards. Run bounties and rifts as often as possible, then clear all your shards on 1 slot (ie head, shoulders, etc) at a time until you get a set piece. Keep moving from slot to slot until you have a 6 piece bonus. That alone should help you farm faster and get to higher rifts.
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Another way to quickly get some of the tier pieces is Blood Shards. Run bounties and rifts as often as possible, then clear all your shards on 1 slot (ie head, shoulders, etc) at a time until you get a set piece. Keep moving from slot to slot until you have a 6 piece bonus. That alone should help you farm faster and get to higher rifts.