Steal the dagger from Enchantress or Spear from the Templar when you start the game, they have a much better weapon than your basic wand. If you start in Act 1 you can also quickly grab the blacksmiths book and the one just inside the door at the Inn, they give a bubble of experience per book. This isn't necessary though. Few ways after that you can go (rifts, bounties, etc), but you should check the vendors at 7 and 11 for rings and amulets respectively for x-y damage stuff as they'll massively increase your damage. After that you just rift.
You can also just do bounties until 11 and clear a full act. Leave the hand in until your level 23 though. You'll get a bunch of crafting recipes from the bag and ideally you'll get either/both of Cains and Borns which have +% experience bonuses but you'll only get that at 23+. If you do that route make sure you do the "Bonus" act for the extra bag.
Also when you're about to level, try to grab a huge chunk of monsters then ding yourself. The AoE damage you do when you level is massive and can obliterate a champion pack which is extremely helpful for picking up the pace. Same as if you do a boss bounty and you're getting near a level. If the boss has an add like Magdha kill an add near her and take 30%~ of her health off although this is only really effective for boss bounties at lower levels as there generally won't be enough adds to level you up when you're higher level.