@Daniel Lang
Everything is state driven, but in general the leases are turned over to the new owner. An estoppel certificate is given to the buyer at closing that is a legal document that states you are the new owner. You can use this when you talk or notify tenants of the changes of ownership. You should request for a rent roll as part of your due diligence and a copy of all leases. Make sure tenants know where rent payments need to go. That is critical. I always had the existing tenants fill out an application so I have emergency information and other important info. At closing, make sure rent payments credited to you and security deposits.
As @Patrick Menefee stated, security deposits should be held in a separate account. Again, check with your states rules. Could be county or city rules as well. It depends.
The bank accounts can be separate for each property. I started that way, but eventually it became to many accounts. I now use one account and it is controlled through my accountant and quickbooks.
As far as books; Managing rental properties by Brandon, cash flow quadrant by Richard Kiyosaki, profit first by Mike Michalowicz.
Best of Luck.