Tony,
What you are calling the "first action guy" is always the guy to bet first for the entire hand. Bets are resolved clockwise around the table.
If someone raises, the total of previous bets and raises becomes the new bet amount to the player next to bet.
Betting continues clockwise until everyone has called or folded.
For example, let's imagine a 6 player game with Dealer (D), Big and Small blinds (B & S) and players 1, 2, 3.
Case 1: Nobody bets and everyone stays in the game. Well, actually, you need to bet at least the big blind amount to stay in. So, player 1 bets the big blind, followed by players 2, 3, D, and then S puts in the other half of the blind (he needs to put in the other half to call).
Case 2: Someone raises, lets say player 2. 1 bets the blind amount, 2 raises (let's say making the bet 2 times the blind amount). The bet (2x blind) is now to player 3. He calls with 2x blind. Then D, S, B all need to call by betting 2xblind. In the case of the S and B players, they only need to put in the difference from the bet and their blinds. Finally, 1 calls by putting in the raised amount. If all players stay in the game, all will have put a total of 2xblind into the pot. The next round, betting will commence with player 1, the "first action guy".
After the hand is played, the dealer positon moves clockwise (along with all other positions) and then player 2 will be player 1 and begin all betting for that hand.
So, betting always progresses clockwise and just keeps going round as long as it needs to to get all of the raises taken care of.
Does that help? Do you have questions about the limits or only no-limit play? Let me know. I'm no Hold 'Em expert, but I've played a couple tournaments and play "for fun".
Oh, the button usually represents the player who is the "dealer" in a house-dealt game.
Jim