Adding Permissions via Share Point

I work for consulting company and we are opening new to the 365 Office concept. I have set up a team site for our company were we can all exchange files and have access to whatever folder we are synced to. This is all working fine n' dandy till I started on another companies site. I have set up 1 team site for the small company with multiple folders ranging from administrator's folder to remote site folders. 

Now here is the issue: I would like to be able to control (preferably by group) who has access to parent folders as well as sub-folders. I have make (under users and groups in skydrive site) distribution lists (which typically are for e-mail.) Is there a way to make security groups? I am using the web interface as of right now. I have googled ALOT but Skydrive/Sharepoint/365 seem to change quite often. So far I noticed a very jagged permissions system if you click, "...." in a folder then "Advanced" then monkey around till you find manage permissions. My distribution lists see to show up in this but I have no way of editing individual folders/sub-folders. It just seems like a pain to figure out, I can edit what permissions each distribution list has (full control, edit, write, ect) just as a general permissions for the entire group. How to I apply these permissions with a security group and apply them for parent folders and sub-folders or even how do I control them on a per-person permission basis?

Thanks,

-Michael