From watching videos and going to local Mexican restaurant, it seems the tortillas are much thinner than how I am able to get them when I try to make them at home. Should I be pressing harder all the way down? It feels like there's a limit to how much I can press before it starts sticking to the paper too much, is it possibly a problem with the dough consistency? I use Masa Harina and follow Rick Bayless' video https://www.youtube.com/watch?v=dRwMu9ERCKk 
There are a variety of reasons your tortillas might be thicker than you'd like.  My first tip to you is to line your tortilla press with a thick plastic sheet on each side instead of paper; I use a circle cut from a freezer bag.  This makes it easier to peel thinner tortillas loose. Beyond that, some troubleshooting:  Flour consistency: grainier doughs, such as those made from hand-ground hominy, can't be pressed as thin as ones made from very fine ground masa harina. I've made tortillas from our local Oaxacan market's masa para tamales, and those are pretty chubby because of the coarse grain. Dough consistency: if your dough is too sticky you won't be able to press it thin without it sticking hopelessly.  If it's too dry it'll be too stiff to press thin, or will crack.  It takes some practice to get exactly the right consistency.  That aside: enjoy your thick tortillas!  Those are actually a legit tortilla style, and they help keep your taco from soaking through and coming apart. 