Although Pranav Hosangadi has explained, I will try to explain where you might be going wrong. I think this will be helpful for you. And I think it will not be waste of time in typing the answer for you.
I was going to ask: if mass is an objects tendency to resist acceleration then why do two objects of different masses fall to the Earth at the same acceleration?
Yes, you are right. If we want to push a feather we need less force and the feather resists the acceleration to less extent, and then if you want to push a iron ball we need more force and the iron ball resists the acceleration to more extent.
So, it is correct that mass is the tendency of object to resist acceleration.
Then, why do feather and iron ball gets accelerated to same extent towards the earth?
Isn't iron ball's mass greater than feather's mass, so iron ball should have greater tendency to resist acceleration than feather, thus iron ball should fall first than feather?
Here, if we would had applied same force on iron ball and feather, iron ball would had greater tendency to resist acceleration than feather, so iron ball would get accelerated to less extent than feather. Here, although iron ball gets accelerated to less extent, its greater mass than feather makes force to be equal to force exerted on feather, where acceleration of feather is more but mass is less.
But, earth doesn't exert same force on feather and iron ball. Force exerted on iron ball is greater than force exerted on feather. As it is confirmed from Pranav Hosangdi's answer that acceleration due to gravity is independent of mass of the body to which earth exerts force.
Then I read those posts and it seems that even though it is very small, the more massive object falls faster. Okay I understand, both objects attract each other.
Seeing your comment, I hope you have understood that because of air resistance iron ball falls first than feather.
If two cars of different masses collide doesn't the car with less mass accelerate more even though both cars received the same Force.
Yes, the car with less mass accelerates to greater extent than the car with greater mass as you stated force to be same on both.
Then that implies you need more force to accelerate a large mass than to accelerate a small mass. Because that is how I see it, the Moon attracts the Earth with the same Force as the Earth attracts the Moon but the Earth accelerates less due to its larger mass.
Yes, you need more force to accelerate a large mass than to accelerate a small mass. That is the reason earth exerts more force on massive objects than on tiny objects. You are also correct in saying about earth moon interaction.
So then how is mass an object's tendency to resist acceleration?
So far there is nothing which has contradicted with how mass can't be tendency to resist acceleration. If there is any thing you can comment.
. So should we not really be able to see the difference in acceleration when dropping a massive object?
Massive object and tiny objects are not exerted with the same force by the earth, you are thinking that on both the object force is same. Earth exerts different force on massive objects and tiny objects. Acceleration due to gravity has constant value at different points in space, just like electric field has constant value at different points in space. Acceleration due to gravity can be considered as gravitational field. So, no matter what mass you keep, it will be accelerated to same extent.