if you look at that zone list, there's basically 3 paths for 1-60 (more like 1-58 but whatever) through old world stuff.
Horde has 2 paths through Kalimdor, and 1 through Eastern Kingdoms. Alliance has 1 through Kalimdor, and 2 through Eastern Kingdoms.
The start of horde Kalimdor path 1 is Azshara. Path 2 starts in North Barrens, path 3 starts in Silverpine.
Alliance has path 1 starting in Westfall, path 2 starting in Loch Modan, and path 3 starting in Darkshore.
There's still basically a unique starting zone for each race, although the shared areas (gnome/dwarf and orc/troll) are still shared from about level 5 on. Gnomes start out in the reclaimed gnomeregan, trolls start out in the reclaimed echo isles, and then at level 4/5 they go to the same starting town as the dwarves/orcs. So I'm listing these paths as starting with the level 10 zones where they go in different directions. You aren't locked in to following the natural zone progression, but each place is set up so you will be funneled into a new zone right next to the zone you just finished.
and I should point out that you actually can't progress some of the story quests for zones without doing other zones first. It won't affect anyone leveling a new character, but like if you have a level 80 and you decide "hey I want to check out the new stonetalon" you might have to do some quest chains in ashenvale first.