Send her a 'message' through someone if you 'want to know her more closely'. If her response is "OK" you can start a Ta'aruf.
Ta'aruf is a process when a man and a woman with intention to marry try to know each other closer before they decide to marry. You must visit her home and do silaturahim with her family in this process. You can talk with her, but of course her mahram must present.
If in this ta'aruf process you and her feel 'matched', you can go the next process, Khitbah. However, if you and her feel cannot match, you must stop the ta'aruf process.
Khitbah is a process when a man and a woman start to talk seriously about a marriage such as what the man expect from a wife, what the woman expect from a husband, where they will live, etc.
The next process is nikah.
But actually the ta'aruf is not always necessary. A man can propose a marriage and start the khitbah if the woman accept this proposal.
That's the halal manner that you can do if you have been ready to marry someone. But bro, you said "First off id like to say im only 19, marriage is certainly not on my mind", it seem like you haven't ready to marry. So, don't approach her for this time.
If she is your wife in the future, one day you will marry her. But if she is not your wife in the future, InshaAllah you will meet another woman and your feeling to your classmate will gone with the wind.