Before you can begin to mix like a DJ, you must think (and listen) like one. Spend time sampling a variety of songs by different artists. Pay attention to tempo. Count the number of beats in your head, starting over with each new verse. As you broaden your palette, you will begin to enjoy songs for their "mechanics" (how well you can identify their makeup) rather than simply their sound or lyrics.
Set aside a few songs with easily recognized bass patterns. Patterns are measured by how many times a beat plays in a song segment before repeating. Those with 4 definitive beats that repeat often are easier manipulated than those with 8 or 16. "Jump Around" from "House of Pain" is a classic example of a 4-beat pattern, compared to Run DMC's 8-beats in "Walk This Way."
Listen to two of your favorites out of these songs several times over until you can count the beats in your head. Mentally divide the songs into these parts: the intro, the verses, the chorus, the bridge and the "outro." Most of your mixing will be centered around the intro and the chorus, so consider how those of one song might sound when coupled to those in another. For your first attempt, try to choose songs with vocals that don't overlap the beats where they repeat.
Find a clean place to start early on in the track where a beat drops and no vocals or other sounds are in the way. Use the jog wheel to set the cue point on one of your decks and press play. Find a pattern of 4 beats (or more) and activate the looping feature on your hardware or software by pressing "in" when the first beat lands. Press "out" at the end of the pattern. The segment you chose should be playing seamlessly.
Now bring in a second song on your other deck the same way. You can either create another loop or play the new song through over the loop you already set. Chances are the songs aren't perfectly in sync, so adjust the pitch slide on the slower of the two tracks, raising it until the patterns synchronize on both tracks. This is called beat matching.
Once you've conquered matching beats and manage a clean mix, fade out of the earlier track using the crossfader and create another loop on the fly. As that track repeats, cue up another song on the opposite deck and unveil it the way you did in the previous step. With practice, you will soon notice your beats syncing up better and better with less effort.