I'd recommend the Barclay's Arrival card with it's $444 signup bonus when redeemed on travel. It's not a traditional miles card and works more like a 2.2% cash back card. You receive "miles" for every dollar you spend. When you want to redeem your miles, you are presented with all of your recent purchases coded as travel expenses and can choose to reimburse yourself.
I just started dabbling with travel hacking recently, and got started with milevalue.com. I second the Barclay Arrival.
My wife and I are flying to the Grand Canyon in April (hiking to the bottom, camping two nights, hiking back up). I priced out the air fair ahead of time and it was going to cost us about $950-$1,100, ouch! So, I learned a little bit about that travel hacking thing I had heard of, and got both a Frontier Airlines card and a Barclay Arrival. At the end of the day, we're flying to the grand canyon for a total of $80, and most of that is a (very worth it) annual fee on the Frontier card.
Note that they just changed the sign up bonus on the Barclay Arrival card. Originally it was $444 cash back after spending $1000 in the first 3 months (hello 44% cash back!), but they just recently changed it to required $3000 spent in the first few months.