At around level 20, things start to get really interesting. A lot of classes gain some particularly exciting new abilities at level 20, such as the druid's catform or the shaman's ghost wolf. Most of these will become invaluable for the remainder of your levelling time.
Also, at around level 20, it's time to move on from the 'starter' areas. There are several places to quest, with plenty of variety, though you may find quests more spread out than previously. Also, if you're on a PvP server, things can start getting nasty as you venture into contested territories and zones with towns for both factions. Unfortunately, ganking (being killed by higher-level players) is the order of the day -- grit your teeth, stay off the roads, and you'll soon be big enough to visit some revenge on your tormentors.
If you're not bored of the Barrens yet, Camp Taurajo (in the southern half of the zone) will get you quested up during your early twenties. Sun Rock Retreat in the Stonetalon Mountains is an obvious next step from the Barrens, and a level 20 quest from Ratchet sends you to the zone to hunt spider eggs. However, most of the quests from Sun Rock Retreat are around level 26-27, so you will probably want to forget the spider eggs until you're ready to tackle the entire zone.
Another excursion from the Barrens takes you to its northern neighbour Ashenvale; this night elf zone stands in contrast to the arid Barrens. With two Horde bases (Splintertree Post and the Zoram'Gar Outpost), there are a few quests in this zone (most suitable for those past level 25).
If you're not on a PvP server, you may find the Hillsbrad Foothills a zone worth adventuring in. Horde town Tarren Mill contains a number of quests, but nearby Alliance town Southshore is frequented by higher-level players, so the area is PvP-heavy. Hillsbrad is a natural progression from the earlier undead zones.
Towards the end of the 20-30 bracket, you can venture into Thousand Needles, south of the Barrens. Quests abound at Freewind Post, and the neutral quests in the Shimmering Flats (southeast of the zone) can be attempted around level 30 (the mobs are level 30-35) for a lot of solid XP all at once. The Flats are large, and there's no Alliance town to encourage PvP.
Players adventuring in Westfall will find it easy to continue their adventures in Redridge Mountains, though several of the Redridge quests can be done in your late teens. Darkshire, in Duskwood, is the next step -- you'll be travelling between Raven Hill cemetery and Darkshire a lot, but the quests here stack very well, meaning you can get several done at once. You can spend most of your time from 20-30 here if you choose.
In Kalimdor, Astranaar in Ashenvale Forest is the next night elf town along from Darkshore, with a few quests for the early-to-mid twenties. On PvP servers, you may find war-happy Horde players launching raids against the town -- the zone is large and questing mostly takes you away from the action.
The Wetlands town of Menethil Harbour is also worth a trip, with tons of stackable quests (and it's not far from Ironforge). Spend time in one or both of these zones before heading back to Redridge and Darkshire to finish up the harder quests -- then you can visit the Stonetalon Mountains for your late twenties, and, as with the Horde, make a start on the Shimmering Flats' quests.
Blackfathom Deeps (BFD)
If you're over in Ashenvale, this instance is for you, with quests for both Horde and Alliance. It's eminently skippable but a fair few early-20s blues drop, and sizeable XP can be obtained from the quests.
Shadowfang Keep (SFK)
A Horde-friendly instance in Tirisfal that has some notable rare drops, including the twink-friendly Assassin's Blade and Shadowfang; it may be worth the trip for Alliance players, or you might prefer to solo it later and resell the loot.
The Deadmines (sometimes called DM)
If you haven't already finished the Deadmines quests and walked off with some nice loot, now's the time. Horde players have a long way to travel and no quests here, but some think the blue drops are worth it.
Similar to RFC, this instance is inside a city -- Stormwind. It's fairly easy (and some say boring), but if you're lucky the rare spawn Bruegal Ironknuckle will be visiting and drop a Prison Shank. Few other blue items drop, so if you're after loot, you might be disappointed.
Again, you can gain some XP (and, of course, honour) from PvP. As the rank you can obtain is limited by level, it's harder to get a good rank at this stage, and the XP is slow -- partake in PvP for fun rather than for levelling.
[Note: I apologise for the (several months') delay in this series. Levelling alts takes time!]