Hooray. Outland. Now this is the real home stretch. This, at least for now, is where most of your adventures will happen. After slogging through the first 20 levels, grinding up to the landmark 40, surviving the boring trek to 50, and eventually making it to The Burning Crusade content, the time has finally come to make that final push. You should probably celebrate a little, because from here on forth you will get new trainable abilities at every level so don't forget to pay your trainers a visit. Although odd-numbered levels usually have higher ranks of old, little-used spells, so it's not a huge deal to skip training between levels. Besides, the first few levels in Outland are such a breeze that you can hit a few levels without getting a chance to visit the old world.
By this time, you really should be riding around on your pimpin' new mount. For Horde players, it's extra special because the Blood Knights get a tabard that's second in coolness only to the Tabard of the Shattered Sun, so there's every reason to complete the quest chain. If you entered Outland at level 58, questing in Hellfire Peninsula should get you past 60 in a very short time. The experience gains from quests are vastly superior to the quests in the old world, as well as gives heftier Gold rewards. This is important because you should be saving your money as early as now (if not sooner) in order to afford flight training.