Don't use chmod 777, afaik it might cause some problems.
A better option would be to use the chown command to change the ownership of the /mnt/somethingSomething partition and make it belong to your user and group (which, in turn, will make your user have read / write / execute permissions in that partition that he owns).
It goes like this chown -R user:group /mnt/PathToThePartition
To find out your user name and group, use these commands whoami and groups - to respectively find out the user and the group. On my system the group is named the same as the user.
Eg, If your user was named Steven_Self, and your group was called Steven_Self, it'd be chown -R Steven_Self:Steven_Self /mnt/PathToThePartition
As to a separate home partition, it's possible to use it, but you'd need to reinstall the system to do it, I don't think there's a way to transfer an existing home directory to a new /home partition.
To do it in the installer, choose the manual partitioning option, create a separate partition for /home and give it the home flag (you can add the flag via GParted).
