Cable TV only provides you with TV channels.
If you want to get internet service, then you need to contact your cable provider and inform them you would also like internet cable service as well. There are generally different price packages for cable internet service. Low price packages tends to restrict how fast you can download and perhaps how much bandwidth you have (the amount of total data per month). High price packages then to offer the fastest download speeds and as well as faster upload speeds. Uploading is always slower than downloading. There should also be a "standard package" which will typically give you unlimited bandwidth (depending on your provider), good download speeds and slow upload speeds.
The internet service provider (ISP) will basically rent you to hardware to connect to the internet. That include the modem and the router. Sometime it is a single unit other times it might be two separate units. If possible you should purchase your own equipment rather than rent them from the provider. For example, I purchased the modem myself which cost me around $70 because I didn't not want to pay $5 per month to rent the modem. It takes 14 months to breakeven, and every month afterwards I basically save $5 + the taxes changed on that $5. The downside is that if the modem breaks I need to buy another one whereas if you rented it, the ISP while give you another one.
The router allows multiple devices to be connected to the modem and it supports WiFi connections as well to the modem. Since I already have my own wireless modem, I didn't have to bother with renting one from my IPS. The modem typically only has one port for a device to be connected which is why it is generally recommended to connect it to a router. If you only use one device, then it's not a big deal. Since you have a laptop I would assume you prefer not to connect it to the modem with an ethernet cable so getting a wireless router would be a better option.