Well, grams are what's important in the macroscopic world of the humans, so we want to discuss masses of things in grams.
However, if one were to label the masses of the various elements of the periodic table in terms of their weight in grams, there would be a nasty 10^{-23} factor on all of the elements because they are so tiny. Thus we use atomic units for the weight of the individual atoms on the periodic table (this way weights are something like "10.5 amu" or "212.4 amu", instead of "3.42x10^{-23} g". Atomic mass units are much more in the ballpark. The conversion rate between grams and atomic mass units we call Avogadro's number. There are 6.022x10^{23} atomic mass units in one gram.
Now, when working with elements in the lab, we often want to measure out a certain number of individual atoms of some element. Of course, we want to measure out some large amount that we can work with, any small amount (less than a gram) is very hard to work with in the lab. How do we know exactly how many atoms are in some arbitrary amount of grams of a set substance? Well, we can convert from the number of grams of the substance to how many atomic mass units of the substance we have using Avogadro's number. Now we can just divide our total amount of atomic mass units by the amount of atomic mass units the periodic table says is in one atom of the element. We now know how many atoms of the substance we are working with. Thus we can now easily match up an equal number of atoms in one substance to another to do some particular reaction even though we are working with huge amounts of the substances because we don't like to work in anything other than grams.
"Moles" and "Molar mass" are just analogous to the lingo that one would see on the periodic table where we are working in atomic mass units, but instead it's the lingo we use for the macroscopic (gram) world. Molar mass is like the atomic mass, but instead of how many atomic mass units are in 1 atom (atomic mass), molar mass is how many grams are in 1 mole of a substance. We can not work with single atoms in the lab - we can work with moles of a substance (half a mole, 1 mole, 3 moles, etc.), because they are large amounts of atoms such that a mole is in the ballpark of being a gram, tens of grams, etc., which is something we can work with in the lab yet still retain the exact information of how many atoms we are working with so that we can correctly match two substances together atom for atom so that we can perform some chemical reaction and know exactly how many reactants are left over (perhaps we don't have equal amounts of atoms (and therefore moles) of the reactants in a reaction.)
I still haven't said how many atoms are in a mole, though I did mention in the last paragraph that a mole is an amount of some substance (I DID give you what that amount was in grams - it's the molar mass). Here Avogadro's number shows up again, this is simply because it would be nice if in the case of Carbon, where there are 12 amu per atom, we could have something similar like that in the macroscopic world of grams. Thus we want 12 grams to be 1 mole, so that everything in the microscopic periodic table world is analogous to everything in the macroscopic gram world. It would be stupid to have 12 amu per atom of Carbon, and then go and have 55 g per mole of Carbon. If we use the same conversion rate for amu to grams as we do atoms to moles we are in this analogous situation where 12 amu per atom of Carbon means that a mole of Carbon is also 12 (grams). This choice of the amount of atoms that a mole actually is is also convenient in that it makes a mole of atoms convenient to work with in the lab, because we like to work in grams, tens of grams, etc.
In summary, these quantities are all mentioned early in chemistry courses because we can't work in the theoretical framework of chemical equations involving 4 or 5 different atoms when we actually go into the lab and try to perform these reactions. We don't talk about 4 or 5 atoms in the reaction, we say there are 4 or 5 moles in this reaction, we have converted all of the language one would use with individual atoms in a theoretical context into practical quantities of the atoms such that we can perform the chemical reactions in the lab with this new language for the quantities of the entities involved in some particular chemical reaction.