Well, the DMG and Player's Handbook will answer all of your questions for you but, I'll simplify what I can:
Let's start with attacks:
Whenever a Player Character wants to attack or you want to attack a PC with a monster, you have to roll to beat one of their 4 defenses (Armor Class, Fortitude, Reflex or Will; the attack will usually note which attack is against which defense). You'll roll your D20 and add the necessary numbers to the roll. So, for example:
Urrim Stonereaver is a level 1 Dwarf Fighter (
http://mydndgame.com/character/2602/sheet) and wants to attack the Deathlock Wight the group is fighting with his Tide of Iron power. He rolls his D20 and gets an 12, Urrim's player now adds the Proficiency of his Battleaxe (+2), his Dwarven Weapon Training feat (giving him a +2 with attacks using axes) and his Strength modifier, the attribute used for Tide of Iron.
The total result is a 19 (12 roll, +2 Prof, +2 feat, +3 strength). Looking at the Deathlock Wight's AC (18) Urrim's attack succeeds and he rolls damage as per Tide of Iron's description, doing a total of 1d10+5 damage (+3 Strength mod, +2 from Dwarven Weapon Training) and pushing the Wight one square back while Urrim steps into the Wight's previously occupied square.
Attacks agianst Fortitude, Reflex and Will work much the same way in that you look at the power's description (for example, a Warlock's Hellish Rebuke uses Con vs. Reflex) and roll based on that. Implements won't give proficiency bonuses unless they are enchanted however.
Since you're DMing though, the Monster Manual will have the numbers you add to your roll already calculated for you (for example, the Deathlock Wight's Grave Bolt power is laid out as +6 vs. reflex) and you just need to roll agianst the target's defense.
Now, as for attacks of opportunity:
Any player (or monster) gets an opportunity attack if an enemy within their reach (typically 1 square for PCs, some times more for monsters) and moves more than 1 square (which is considered shifting), uses a ranged attack or an area attack. Any threatening player or monster can make a single Melee Basic Attack (there will be some powers that can be used as melee basic attacks, so be ready for that) against the opponent. And yes, this action is done during the opponent's turn. So, let's use Urrim and a Deathlock Wight for an example:
The Deathlock Wight attacks Urrim with Grave Bolt, which is a ranged attack. Urrim gets an opportunity attack however because he was able move up after pushing the Wight back. So he makes a basic melee attack (1D20+7 for the attack roll and 1d10+5 damage if he hits) and deals his damage. If this attack brings the Wight down to 0 HP, it dies without resolving its attack. Otherwise, the Wight finishes its turn as normal.
Page 290 in the PHB has this detailed with a bit more information. I don't know if that's where you were looking, but definitly look at it if you haven't.
If you have any other questions about DMing, feel free to ask and I'll answer them if I can. And good luck with your new campaign
