The picture below sums up everything you need to know, but I'll explain it a bit further than that anyway.
There aren't many different core ways to build your team. Even though they are all effective, the one above is the easiest to use.
Here are the most common ways to kill an enemy: destroying the tank, sniping out or sneaking. You can only progress if you protect yourself from all of these 3 strategies, every time.
Your tank must have shield, heals, void, high hp, armor, or whatever else you can use to make him last for the entire battle. He should not die. It is very important you choose something that doesn't die.
In the picture above, I take advantage of an equalizer match to use a monster usually killed easily by magic attacks. Because Beetle Queen will heal him for more thanks to his high equalized hp, he won't have any problems surviving.
Many people use Cerberus, but Cerberus isn't that good in high mana matches because he can get taken out by direct attack strategies. I.e.: water splinter can destroy Cerberus by using high speed, magic monsters to kill him before he can heal himself. If he can't survive, then he's a bad pick!
Direct attacks can use blast. That means your 2nd monster is also in danger of being taken out by melee attacks! You can't use a ranged monster here because ranged monsters are very fragile. Besides, if you don't have an off-tank and your tank dies, then you lost. An off-tank counters a strategy focused on destroying your tank.
There's something else the off-tank needs to do. It has to block snipe attacks. If you don't have a monster to block snipes, your support/damage monsters will die easily in case your opponent has a good defense focused on snipers. Your healer will die easily if it gets focused. You can't let that happen, so use an off-tank.
Damage / support
Your best monsters go here. They don't have to do something specific, They just have to be the best you have available for that match. When you don't have a healer or the match doesn't allow healing, you fill in with damage or repair or whatever else you have at the highest possible monster level.
Fly damage / fly support
When I use the word “fly”, what I really mean is anti-sneak. There is no sneak attack monster in the game with the flying ability. Using something with fly for your last monster increases his chance of surviving longer than any other monster, simply thanks to luck.
Other abilities you can consider instead of “fly” are: shield, dodge, heal. Whatever makes your last monster more likely to survive an onslaught of sneak attackers.
Wait, what about opportunity?
Opportunity attackers build upon whatever weak spot you have. If you have none, then you are already doing the best you can. If you're worried about opportunity, congratulations! You already know the basics. Only experience and further analysis of your own games will improve your skills.
Monster function examples
Maybe you're not sure of what a certain monster is best at doing. I'm not going to post an entire list of them, but here are some low-level card examples:
I have X'd out the ones that are too weak to consider while compared to that collection's cards available.