Different people will have different ways of approaching no limit Texas hold 'em strategy. While there are many different ways in which strategy can be approached, giving you a very small beginner strategy to follow that will do a lot for keeping you out of trouble is a relatively simple matter.
There are a number of different starting hands in hold 'em, but only some of them are actually worth playing. Here is a rundown of the different starting hands and what you should be doing with them when you get them.
Huge Pairs: AA and KK. When you have these hands, raise! If you get re-raised, then raise it again! Do not be afraid to get all of the money in the pot with these hands. Dan Harrington says that you should not be worried about getting it all into the pot with two Kings, specifically because you will be net plus in the long run and even if your opponent has Aces you'll suck out around 20% of the time.
Big Pairs: QQ, JJ and TT. Play these hands aggressively as well, but be prepared to lay them down in the face of heavy raising and especially with heavy raising when there is more than one opponent doing the raising. With two opponents constantly raising each other pre-flop, the chances that one of them has Kings or Aces is just too great for you to continue with this hand.
Medium Pairs: 99, 88, 77 and 66. If you are in early position and the game is pretty tight, consider folding these hands. If you are in early position with a relatively loose-passive set of opponents, then consider limping in. From late position, consider raising with these hands.
Small Pairs: 55, 44, 33 and 22. Do not play these hands in early position; it will only turn out poorly for you later on. In middle and late position you can consider calling with them if there are going to be a lot of other players in the pot. You want to make sure you have the pot odds to call (especially from middle position). Do not be afraid to attempt a steal with these hands from the button or the cut-off if nobody else has come in before you.
Random Big Cards: Anything from AK all the way down to JT. These cards are the lifeblood of the typical no limit hold 'em player because they are going to be the cards you are in the pot with most frequently. The higher the cards, the earlier you can call and raise from in terms of position. AK down to QJ can be played in early position, with everything else playable in middle and late position. Definitely raise with Aces.
Other Cards: The only other cards you should really be considering playing as a beginner are the suited connectors; T9 suited down to 54 suited. Try not to play anything below 54 suited because 54 is the lowest suited connector that can form the maximum number of straights (three above down to three below).