SAP software is huge and there are so many different areas into it. It is best to undergo training to get started on SAP. Since your company is going to implement SAP soon, most likely you will be imparted training depending upon your role. But if you want to take initiative and have ambition to get into an SAP role, you may need to put extra effort.
SAP Education (a division of SAP) and SAP Education Partners (e.g. Siemens Information Services, NIIT and a lot many) provide trainings - classroom, virtual and e-learning. But that may cost you a lot. There are several online training providers who provide SAP training at reasonable price. You may Google and find a good one. The SAP forums may also help you in your search. You already have SAP system access, so you are spared from sending money on this front. Otherwise it too may cost you depending on your usage.
Still if you want to look for resources which can help you do things on SAP system, all I can tell you is that there is no such organized SAP Handbook. You may go through a book on using SAP and for general information. Yes, for different topics you will get tutorials which guide you step-wise-step. For example, suppose you want know how for LSMW (Legacy System Migration Workbench) which uploads data from your local system to SAP system so that SAP users can make use of it. You can search for LSMW and you will find information (e.g I found this in Google search results: LSMW FAQs) which will help you to carry out the task. The point I want to make is you need to know what you want to do and then you can search for the required information.
Probably in ABAP (SAP Programming language), it's possible to get started. You can go through an SAP Press Book "ABAP Objects" from Horst Keller and Sascha Kruger which starts you with your first ABAP program.
If you want an overview of SAP software and various solutions in the mySAP Business Suite family (which you will come across while implementation and after-wards at your company), the following series of articles may help you:
Hope it helps:-)
CHEERS ...