The thrill of meeting someone new and the subsequent mutual attraction is quite a powerful one. Appetites wane, sleep often takes a backseat and you can't get your new special someone off your mind. During this stage, you may find that you are more spontaneous and trying things you never have, such as staying out at an all-night diner, gazing into each other's eyes. This stage can fade quickly, however.
Involvement occurs once the initial rush of meeting someone new subsides a bit. This is the point where you begin to learn the basics about each other, such as middle names and favorite ice cream flavors. On the flip side, this is also a time where future pet peeves can manifest themselves. Initially, something like habitual tardiness may seem almost cute, but it may very well drive you crazy down the road.
This is the point when things become official. Intimacy, both the physical and emotional kind, is established and the course of the relationship is set. Prior to this stage, a common worry is that even though things are going well, the other may just see it as "having fun." Once exclusivity is set, however, the initial excitement makes a return, as it is confirmed that the person you are dating feels exactly the same way you do.
Regarding your first argument, it is not a matter of if, but a matter of when. This is a pivotal stage, and how you both handle it will foretell a lot about the strength of the relationship. The cause could be anything: a misunderstood gesture, a forgotten date or even jealousy. You can learn just as much about each other during the rough times as you do when things are going well.
Once you've weathered a few storms together, your relationship will begin to feel like second nature. You now see your lover as a friend as well, and you support each other on all levels. There is a fine line between comfort and boredom, so it's important to make an effort to keep things interesting.