As a Beastmen player who played Dwarfs for several years before, I can tell you that beyond a doubt one of the biggest downfalls of the Beastmen army is low leadership. You will have a hard time counting on your beasties to do anything that you want them to if there's a Leadership test involved. Here are the main ways that I mitigate the Leadership issue:
One important thing to remember about this army is that the characters have to support the troops with Leadership. The low base Leadership of the troops means that without characters, your troops will easily panic or break from combat even if they only lose by a few. One of keys to playing this army well is learning where your characters will be most effective, and keeping them there.
This is one of the best benefits to the Beast Army. Movement 5 at a minimum on everyone is great! Years of playing Dwarfs taught me one thing: reactionary movement is the only movement necessary. That definitely does not hold true for Beastmen. It's great to make the enemy have to react to what I'm doing, and what they think I'm going to do next turn. The availability of Harpies and the Jabberslythe give some serious flying capabilities. The Harpies can be used to take charges so you can control which enemy units end up where, hunt war machines, or just hover around as an uncommitted threat. The Jabberslythe can do the second and third of those things as well, plus potentially kill off low Leadership enemies using the Aura of Madness. If you feel the need for more flying goodness, a Doombull or Gorebull on a Flying Carpet can pack a real punch.
Ambush is another example of mobility. It is a a double-edged sword, of course. Your units
may never show up! When they do though, you have a decent chance of having them show on
either flank, or in the enemy's backfield. All of which are decent options, and the fact
that you can start testing on turn 1 is huge. Some players will do bizarre things when they
know there will be a unit or two appearing behind their lines, or on their flanks; those
bizarre things can often give you a leg up in the game. Even if they keep a level head,
they'll still have to adapt their plan to account for your Ambushers. A decent sized unit
of Gors or even Ungors can wreak some havoc when they pop-on next to or behind the enemy.
Breaking up your opponents battle plan is always a good idea!
That said, I don't recommend Ambushing. Having tried it many times, I have rarely had any luck with it. There is a 1/3rd chance of your unit not showing up each turn, and a 1/6th chance of your opponent getting to dictate where your unit goes. That leaves a 50/50 chance of getting to drop your unit somewhere that might be useful. Even then, the flank results may or may not be ideal should they be rolled. Ambushing also breaks one of the most important rules of the Beastmen that was mentioned above: The characters have to support the troops. When you have a successful Ambush, you end up with an unsupported isolated regiment. Many times I have brought an Ambushing herd on, just to have the enemy turn all shooting and magic at it; the obvious result being that they panic right back off the table.
Try Ambushing a few times in friendly games, just to get a feel for it. I wouldn't go near it in a tournament/competitive environment if you want to succeed.
This has been one of the best things about Beastmen. You can make a very fast & jukey list to run around, and cause trouble for your opponent. You can make a list full of big monsters and hard hitters to pummel your opponent into the dirt. You can go super magicky, or super fighty with characters. There are so many possibilities! So far what's appealed most to me is a mix of large blocks of Gors, Bestigor, Minos, and even Ungor. I've been using a Jabberslythe, and having a lot of fun with it (though he is far from an optimal choice). These things make the Beastmen army a lot of fun to play with, and keep me from getting bored with it. Also, this has led to many great modeling/conversion opportunities.