Calculation of molarity using dilution equation Using the formula $C_1V_1 = C_2V_2$ I am supposed to calculate the molarity of dilutions. 

The original solution is $\pu{0.100 M}$ at $\pu{10 mL}$. I diluted it by mixing a $\pu{7.50 mL}$ stock solution with $\pu{2.50 mL}$ water, and made another one by mixing a $\pu{5 mL}$ stock solution with $\pu{5 mL}$ water.     

Intuitively, I should get a higher molarity for my first dilution since it is higher in concentration, but when I calculate using 
$$0.1(10)=C_2(7.5)$$
for the first dilution and
$$0.1(10)=C_2(5)$$
for the second, the second dilution's molarity is higher than the first. I know I am doing something wrong but I just can't figure out what.
You're mixing up the terms and formulae. Instead of rote learning the formulae, remember that the moles of the solute will remain the same in a solution - before and after dilution.

With that in mind, let's inspect your $\pu{7.5ml}$ stock solution. Before dilution, it had $\pu{7.5ml}\times\pu{0.1M}$ moles of the solute. After dilution, it has an unknown molarity. Let that molarity be $x ~\pu{M}$. Then, the new moles are $\pu{10ml}\times x~\pu{M}$. Equating and solving, you get $x=0.075$.

If you do that again for the second solution (I give it you as a homework), you'll get $x=0.05$. This easily verifies your own intuition that you should get "a higher molarity for the first dilution since it is higher in concentration".