The units' attack and defense values are simply what you need to roll on the die in order to hit with them. Infantry, for example, hit on a 1 when attacking and on a 2 or less when defending.
One die is rolled for each attacking and defending unit in each round of combat. The die results that show numbers less than or equal the units' combat values are hits, and the others are misses. In your example, the attacker would roll three dice for the infantry, looking for 1s, and three dice for the fighter and tanks, looking for 3s or less (fighters and tanks both attack on 3). The defender would roll four dice, looking for 2s or less (infantry and tanks both defend on 2).
First, the attacker rolls his/her dice, then the defender must choose a casualty for each hit rolled. These casualties can generally be any unit the defender desires. Any unit type can hit any other unit type, with certain stated exceptions (subs can't hit air units, for example). So, infantry can indeed hit fighters. The defender's casualties are moved behind the casualty line, then the defender rolls his/her dice (including units behind the casualty line), and the attacker selects casualties for any hits rolled. Both players remove their casualties at that point. The attacker may then choose to press the attack, starting the whole process over again, or retreat. This continues until one side wins, both sides are wiped out, or the attacker retreats.
I've never heard of an "A&A for Dummies" book, but you can check out the FAQ for more information. There are also some good strategy articles at Axis&Allies.org.
Enjoy the game, and please don't hesitate to ask more questions!
"War is much more fun when you're winning!" - General Martok
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