Is there an easy way to remember charges on ions?
I don't believe there is any simple way to determine what they are except by just plain memorization. I have given some tricks below that have helped me.

When I write down my poly-atomic ions, I generally put them in similar groups in a certain order. From least amount of oxygen to most amount oxygen. 


hypo (name of ion) ite -> least oxygen
(name of ion) ite -> one more oxygen than above
(name of ion) ate -> one more oxygen than 'ite'
per (name of ion) ate -> one more oxygen than 'ate'


Understanding the above helps most with 'ite' and 'ate'. 'Ate' generally being 1 more than 'ite' for oxygen.

Then I list them from least charge to most charge. I order them this way:


Nitr ite $\ce{NO2-}$
Nitr ate $\ce{NO3-}$
Sulf ite $\ce{SO3^2-}$
Sulf ate $\ce{SO4^2-}$


You can see that 'ate' has one more oxygen, and that I have listed them in charge order. Sometimes looking at the trends in oxygen and charge will help you memorize. I have had to remember them by writing them down several times and now I am comfortable.

Here is another example: Hypochlorite is 
$\ce{ClO^-}$

So, chlorite should be?


   $\ce{ClO2-}$, because it has one more oxygen, and


Chlorate should be?


   $\ce{ClO3-}$, because it has one more oxygen than the above, and


$\ce{ClO4-}$, will be called?


   Per chlor ate, because it has the maximum amount of oxygen.


The above is not a hard and fast rule. One exception I can think of is manganate and permanganate.

Manganate is $\ce{MnO4^2-}$ but permanganate is $\ce{MnO4^-}$; same amount of oxygen, just a different charge.

Trying to figure out any of the above from a periodic table didn't make a difference to me because my professor tests us without it. Yikes!