The quickest way to get a 1st character to 70 is to have someone power level you. Barring that, as you have finished campaign mode, do bounties and look for dungeon clear bounties in act 1-4. Some are faster than others, but they give large xp bonuses for completion. Do them on as high a level as you can where you can move fairly quickly through the whole process. On act IV there is only possible such bounty (clear Hell Rift). Other notable quick ones include act I: Khazra Den (Field of Misery) or Halls of the Ancients (Festering Woods), act 3: Cryders Outpost, Forward Barracks, Underbridge. Save your key rift fragments you get until you hit 70. Try to learn which bounties are fast and which are not.
To avoid complete boredom (at 70), I often do all act I bounties (for the cache), the Key Wardens on acts I-IV and dungeon clear bounties as listed above. If I feel that the bounties in a particular act are relatively easy, I'll do all 5. Act V bounties are mostly a pain, but YMMV.
Best way to get gear (other than botting) is to do rifts on normal for the higher drop rate and fast clear, or on torment I (to get class sets and certain legendaries) or on the highest torment you can comfortably clear on. Or if you have a lot of key shards, you can do what's called rift it forward. Google that for a good explanation.
