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 needs a different network. There are two network types, CDMA and GSM. They are not compatible or interchangeable.
Now, you 'might', depending on your carrier/service provider and your phones abilities, be able to use the phone as a hotspot and let the tablet connect to the internet through that. It works the same way (on the tablets end) as it would if it was connecting to any WiFi.
If there is something else going on, then please do explain.