I have already looked in the security and even group settings as an admin, but can’t seem to find the right places to correct this. I was told that members in the trust level 0 group that they can only reply to posts, but not make a brand new threads themselves.
Where should I look in order to correct this? Thank you!
Did you check the category permission? Who is allowed to create topics in the category? Understanding groups and category permissions
The Max topics in first day site setting could also prevent new users from creating topics
Also maybe worth double checking the Create topic allowed groups admin setting as well, just in case. 
Understanding groups and category permissions might be where to took.