This is going to be supernooby. I'm very new to Destiny. I rented it, I don't really like it, and I want to beat it and move on. I'm trying to play the story missions, but after completing one, I don't see another new mission. I see level requirements being mentioned when I try to read up about this, but do you seriously have to run around trying to level up just to play the story missions? I appreciate any info.
Yes, you are correct. In order to advance in the campaign, you must also advance in levels. To level up one may play-through the campaign missions to receive xp, play in the crucible, complete bounties and strike missions as well as other ways. Unfortunately this game is less of a rent for the weekend and more of a spend weeks grinding for weapons and armour in raids and weeklies, so if that is not your desired play style than this is probably not the game for you.
It's hard to say without knowing what part you are on. What's the last mission you completed? Do you see and green indicator marks on the planet screen that usually tell you where to go next?
The story is fairly quick to finish in one day and you shouldn't need to worry about leveling up too much in order to complete the story of the game. You will pretty much level up as fast as needed for every following mission until you hit 20 and you're probably done at that point.
The story missions are there pretty much there to get you to level 20 and then you can start doing other things like strikes and raids and crucible to get even better gear that has light levels that get you past level 20 all the way to level 32.