If I'm a first-time home buyer and buying a house that has currenly has tenants on a lease, what precautions should I take to ensure that they move out when lease ends?

The general rule is that you purchase a home "subject to" the lease that is in place from the previous owners.  I think that you realize that.  Unfortunately there are no "guarantees" that the tenants will move at the end of their lease.  That is true for you and for the previous owner. But you should read the lease to make sure that you comply with the terms and if it says to give notice say 30 days before then make sure you do it, in writing and with some additional days for mailing certified mail.  Good luck.