Congratulations. You just found out the disadvantage of splitting up the public transportation by regional transportation authorities, namely, that travel between such regions is more complicated.
Thus, without going too much into detail, the answer to your question is a clear: it depends.
Generally, in order to travel between VRR and VRS, you need tickets for both of them or a ticket sold under the state-wide valid NRW Tariff. Now there are few exceptions:
For travel between either VRR or VRS and regions that are adjacent to the border between them, a single ticket for one of them may suffice. Details are (only in Germany language) available in information booklets: http://www.vrr.de/blaetterkatalog/VRS-VRR_Flyer_2016/blaetterkatalog/index.html
The VRR recently introduced the "EinfachWeiter" ticket, which allows holders of valid-for-the-whole-month tickets to continue their journey into the VRS for a flat fee. As a short-time visitor, that does not exactly help you.
There is also the "AnschlussTicket NRW", which allows you to extend the validity of a VRS or VRR ticket to other places in the state of North-rhine Westfalia outside of the where the ticket you start with is valid. These are for individual trips.
So depending on how much you travel, you can either (a) buy individual tickets for your journeys, or (b) buy a day-long valid ticket for the complete regional transportation in the state of NRW whenever you travel, or (c) buy tickets with timed validity (e.g., for one week) for both the parts of the VRR and VRS that you want to travel in. Which is the cheapest option depends heavily on when and how often you plan to travel.