In many cases, the number of each element in a compound can be fairly easily inferred by the normal oxidation states of the constituent elements. In the case of lithium aluminum hydride, we have lithium, which is almost always at a $\ce{+1}$ oxidation state ($\ce{Li^+}$). Aluminum is typically $\ce{Al^3+}$. Thus, we need $\ce{4}$ hydrogen atoms at a $\ce{-1}$ oxidation state, $\ce{H^1-}$ (i.e. "hydride"). We get $\ce{LiAlH_4}$.
Similarly, magnesium chloride is $\ce{MgCl_2}$ because it can't be anything else. No need to call it magnesium dichloride. (This also gets into the difference in conventions for ionic compounds versus covalent compounds.)
Of course, in other cases it may not be as obvious. What about iron oxide? Well, that's a bit more ambiguous, it could either be $\ce{FeO}$ or $\ce{Fe_2O_3}$. For that reason you're more likely to see iron(II) oxide for the former or iron(III) oxide. Occasionally you'll still see ferrous oxide and ferric oxide, respectively, but that's becoming less common.
As far as the trivial naming schemes, there is a bit of a learning curve, but it really does make things easier. Calling $\ce{H_2O}$ "dihydrogen monoxide" is only useful for trolling people who get scared at anything that sounds "chemical". It's "water", everyone knows water. Even for your original example, $\ce{LiAlH_4}$, Wikipedia says "the preferred IUPAC name" is Lithium tetrahydridoaluminate(III) and the "systematic IUPAC name" is Lithium alumanuide ($\ce{AlH_4^-}$ is apparently alumanuide, I personally did not know that one)
Full systematic names aren't supposed to be "easier". They are supposed to be completely unambiguous. The problem is that in many compounds, especially in organic chemistry, it's not just the number of each elements that are present, it's how those elements are connected, which can get rather complicated. Consider aluminum sulfate: $\ce{Al_2(SO_4)_3}$. Calling it dialuminum disulfide dodecaoxide loses the information on the individual ionic character, never mind the fact that "disulfide" means something else. Meanwhile, "aluminum sulfate" is clear.
For the most part, it really does come done to a matter of practice and exposure. There are a lot of names that exist with a fair bit of historical inertia (bicarbonate comes to mind), but eventually you get comfortable with it. Of course, Google and Wikipedia can come in handy if you see something new, but you'll find yourself resorting to that less and less.
