It depends on how you are trying to get online. If you are using a data plan on the phone, then it would have access, but unless the tablet had a data plan as well, it would not be able to do so.
A router isn't necessary, however a modem is. A router is for adding additional access (for Ethernet access) or to extend range. Where as a modem is what you need to gain access. Obviously it would have to be one that has WiFi built in or using a router to add WiFi.
So unless there is WiFi where you are for your phone or tablet to access, then you would need a data plan. For the tablet to have that, you would need to contact your carrier/service provider and see if your tablet will indeed be able to get its own plan through them, or if it...