Well, i took your advice regarding the last portion of your post. I don't have a "sda1", my flash drive which has Tiny Core installed is the "sdc1". I can't remember if i was in root or just tc when formatting these two harddrives, but before i wasn't having any issues downloading from Firefox and saving on to the harddrives or trying to modify directories or files on the harddrives. I can either download the extensions i need to reformat these two drives again, but this time do them in root, or i can just download a clean version of Tiny Core and see if that works to find the culprit. I'm not sure where i would type in those commands you have listed in your post. I'll keep you up-to-date, thanks.
Update: Well, it certainly doesn't matter if i have a clean install version of Tiny Core because what you mentioned regarding being @root and formatting the harddrive was the cause. Anyhow so i got this clean install of TinyCore 4.7.3 and got Xfe installed and its telling me when i click properties on one of the drives, [Owner] User=root Group=root, and User has Read/Write/Execute checked, Group doesn't have Write checked, but only Read and Execute checked, Others doesn't have Write checked, only Read and Execute, and not sure what Special is. If i try to change any of these settings, Xfe just locks up and i'd have to Xkill it. I'm probably going to try to find which extensions i used to format the drives (Can only remember gparted) and see what i can do for now.
Update: Okay, so i went into Xfe as "superuser" this time and didn't have any issues changing those options. I set User as tc for both drives, and set Group as staff. I'm able to download files and save them on to the harddrives from Firefox now, and i have no issues modifying them from the file explorer. Once again thanks.
Update: So i took a look at the properties for sdc1 (Flash Drive) which i have TinyCore installed and it has the User/Group set as root, but for Group/Others it has all three checked. So i changed the settings for the two harddrives to match my flash drive, kept User/Group as root, and just checked Write for Group and Other. Everything still works fine.