How to make big, fluffy "Mexican-style" flour tortillas (without using any special machinery)?
Authentic flour tortillas use lard.  For an authentic taste, use that, or consider using shortening or butter since they are solid at room temperature like lard.  You also might want to consider increasing the fat in your recipe.  Fat will coat the proteins of your flour and keep the gluten network from forming so easily.

I was also taught when making tortillas to let the dough rest for an hour after kneading and before rolling/pressing.  This helps everything relax.

After cooking each tortilla, put them in a pile with a clean towel over them.  The tortillas will sweat liquid, helping them to soften up.  You can also put them all in a heat-safe bag (paper or microwave safe plastic) for sweating.  When storing your tortillas, make sure to store them in a bag and not in the open air.

If your tortillas are too small, you should be able to roll out a larger piece of dough.  

Part of the texture issues might actually be from you rolling them too thin.  If you are used to working with dough your ideas of thin might be different from the overstated ones found in recipes.  Try rolling to various thicknesses.