For fun: Just play through the game, doing dungeons along the way. XP mostly comes from killing shit, so doing dungeons just adds more content at no penalty. You'll finish the game before 60 anyway.
For the quickest XP and fastest route to decent level 60 gear: Play Act 1 up until the quest step "Talk to Alaric" (I think it's on the "The Broken Sword" quest), which is just after you meet the Scoundrel. Go talk to Alaric, then run into the festering woods until you find the Crypt of the Ancients. Clear the crypt, portal out (doesn't matter if you pick up the quest item or not), leave the game, then resume. Take your portal and you'll be back in a fully popped crypt. The crypt has amazing mob density and a guaranteed rare pack. Rinse and repeat until you are level 58 (59 is marginally better, but you risk dinging 60 by accident on the next step).
Once you hit 58/59, ask someone to host a game on Normal and run to the Butcher, then set up a portal for you. Kill the Butcher for a guaranteed level 58 legendary. Do the same for Belial and Azmodan for another legendary off each. Then play though Act 4, dinging 60 in the process, and kill Diablo at level 60 for another guaranteed legendary. This gives you up to 4 decent legendaries right off the bat at 60. Level 58/59 legendary weapons are worthless compared to 60 weapons, but armor pieces are generally better than 60 rares.