I've heard that the person who files for divorce is the one that pays alimony, is that true?

No - alimony depends on a number of factors, including the length of the marriage.  Also usually the amount is a ratio that is determine at the date when the spousal support is established.

Not true. Alimony is based primarily on economic factors--how much did each spouse earn during marriage? What is the earning potential going forward of each spouse? How long will take a spouse who put his/her career on hold for the sake of the marriage or family to get on his or her economic feet? Etc. Elements of fairness may also be considered, but definitely *not* who filed.

Note that if there are children, then child support will typically be awarded from the non-custodial spouse, to help the spouse with custody take care of the children. Also, the property owned by the couple, including savings, investments, etc., will be divided in some fashion.