If your lease was up, then they will usually raise the rent for renewal if you had received a discount before. Most of the time, apartments give discounts on "market rent" and gradually raise your rent each year as you renew back up to that market rent.
So, if you are given a document to sign and start a new lease, that is your notice. However, a lease renewal notice should have been given to you well in advance of your lease expiration. It sounds like the $30/mo is the rent you will pay if you stay past your lease agreement on a month-to-month contract. The $125/mo (a fairly steep adjustment in my opinion) is what they are currently renting the apartment for.
Based on my experience with apartment complexes, you should always do the following:
-Get everything in writing and read it before signing.
-Count on getting a rent increase upon renewal.
-Expect that they will find a way to keep some or all of your deposit (sad but true).