Hi mbmin, good question !
Ok, I'll respond to this in several parts.
Firstly, the armies in warhammer 7th edition and what you might need to know because you haven't been around for a few years.
Secondly, your forces for small games (which we'll call the 600+900 pt games) and for larger battles (1500+2000 pt)
Thirdly, what you can do to improve your chances of winning - without resorting to loaded dice

Dwarfs
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The are still short, fat and ugly - no change there

They are the most dependable and toughest fighting units in the game, probably the same as you remember them.
They also now have access to the frightening Thorek and his ever present gunline. This is basically a difficult to kill runesmith and anvil that cowardly Dwarf commanders place behind a long line of handgunners, quarrellers and other ranged artillery weapons.
If you have agreed to allow special characters you can expect Thorek and a gunline in a 2000 pt game - he's a very popular choice.
In smaller games your Dwarf opponent will most likely be cowering in the corner of the battle ground in an impregnable castle type of formation. Very tough to break into but a nice ambush target to attack from multiple sides

High Elves
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These pointy ears now have the best array of Magic defence and offence in terms of spellcasters, spells, arcnae items and useful magicky things. 2000pt games often include Teclis (the High Elf "voodoo" master !)
These armies also always take repeater bolt throwers (they're too useful to ignore) irrespective of the pts size of the battle.
They often will take Eagles or Eagle riders against fast moving enemies (for march blocking), esp if they're expecting warmachines (for artillery hunting).
Expect White Lions (move through woods), Swordmasters (elite up close fighters) and, for more defensive armies, Phoenix Guard, difficult to kill.
The whole HE army has been made stronger and more armoured than it used to be. Also, it always strikes first in combat irrespective of initiative and weapons.
Because Dragon mounted characters have been so successful on the tournament circuit everyone seems to be taking them these days so you'll need a "dragon slaying" plan.
Dark Elves
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I don't know enough about them yet (having only played against them twice and only having read the book a couple of times).
However, of the armies you are going to face I'd say this is the most dangerous one for Beasts of Chaos. There's so much shooting and direct offensive magic it's scary. You'll be forced to use ambush to attack from several sides just to stay out of the way of shooting.
Dark elves now hate everyone in the first round of combat (they hate High Elves all the time !) and so they get to re-roll missed hits. They have a lot of good magic items too and their characters are impressive. All round they are a lot more powerful than Beasts in my opinion.
Skaven
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I'm no Skaven expert so it would be best to get someone like Dims (another forum member) to answer. Skaven are a gadget army now, even more so than before I think. They're still a horde army too, ie. buckets of rank and file units / outnumber / flank charges etc.
You should be able to outmaneuver them but you're going to be seriously outnumbered all the time.
Wood Elves
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Another pointy missile threat for your Beasts. These guys have probably now been outclassed in the bow fire department by Dark Elves now (esp if you include not only their repeater bows but also the repeater bolt throwers).
Wood elves can now mess around with tree-hugging all day long by moving the woods around the battlefield while hiding in them (yes, they can opt to add wooded terrain if it doesn't already exist).
Still, Beasts and their skimish stye are ok in Woods so that shouldn't be a problem. Also, you have the might to take on the Treekin and even a Treeman with your hardest troops/characters.
I reckon Wood Elves are probably the least of your worries.
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That's all for now, I have to get back to work. I'll follow this up later.