DB/Directory:
Site Admin -> Server -> Session Handling
At the very top an option to use the database for sessions as opposed to a sessions folder in moodledata. If it's un-checked, your Moodle is using sessions folder in moodledata. From what you've said about admin experience, would recommend leaving it un-checked.
Since it's an issue with multiple browsers, sounds like it might be a network filter if one is on the inside of a private network and the Moodle is remotely hosted (outside private network). If you are inside the private network and this is occuring, then it could be either a browser issue or a network issue.
Have FireFox? Can you install? Try that one!
Also, please see the checklist provided by Chris.
Hard to assist without knowing more specific information: remotely hosted, dedicated, internally hosted, 'flavor' (Linux/Windows/Mac), etc.
'spirit of sharing', Ken