Short answer
In the form editor's Settings > Submission settings, real-time notifications are sent within a few minutes for submissions not flagged as spam. My demo portal's page says they go to the form creator by default; you can also notify the contact owner when a known contact submits, and add users. Spam gets its own daily digest in the Spam monitoring beta.
1. The settings
My demo portal, with one user added.
2. Route leads properly
For assignment, use a workflow. See rotate record to owner.