You cannot have more than one US immigration status at a time, and any nonimmigrant status you have ceases when you leave the US. A nonimmigrant who leaves the US and returns can be readmitted in the same status or a different status depending on the circumstances of the subsequent trip to the US.
You can have more than one nonimmigrant visa reflecting different nonimmigrant categories, however, as it seems you do. A US visa merely allows you to travel to the US in order to apply for admission under the corresponding status.
As a former F-1 nonimmigrant, outside the US and in possession of a valid B-1/B-2 visa, you can travel to the US and apply for admission in B-1 or B-2 status. The answers to your questions, therefore:
Will I be able to just go in using my valid tourist visa now?
Or do I have to do something else?
No, you don't have to do anything else.