I've just started a new hardcore champion, and I am/was facing the same issue. What I did was:
- pickup everything and salvage everything so you have enough mats to use the hope of cain option in the cube. Also dont forget about ''Skill of Nilfur'' to convert set-items. Getting the set complete is more important than having all pieces ancient (at first, obviously)
- do some (split)bounty's and hope for some useful legendary's in the caches
- maybe craft some set legendary's just to have something equipped (sage's is quite nice with the extra Death's Breath option for more Hope of Cain)
- get enough tankyness to join public games and not die, but the not dying part is only important for playing hardcore
- get an idea of what items you need for what spec you want to play and gamble/upgrade wisely according to the d3planner or diablofans page
- run t1 rifts and hope for the best
- and most important: be patient, it will take some time to get things started on a fresh character
Best of luck!