AirPort Time Capsule Connection to Internet
First - download and read the setup guides for your Time Capsule. See help.apple.com for your location and your model of time capsule.

You should aim to replace your current WiFi (from your modem) with a new WiFi network (from the Time Capsule). You can extend your modem's WiFi with your Time Capsule's WiFi but this is usually more complicated and will not improve anything.

Your time capsule should be set to create a WiFi network, select that WiFi network on your Mac Mini and you will be able to use the Time Machine backup function. 

Connect your Time Capsule to your modem router with an ethernet cable so that the Time Capsule can connect to the internet through your telephone company's modem.

You can choose to set your Time Capsule to 'bridge' mode so that the DHCP connection-sharing is handled by your modem router. Or you can set your modem router to bridge mode so that the DHCP connection-sharing is handled by your Time Capsule. You may need to check with your internet provider which method is the best in your area.

The Apple documentation includes guides on how to set it up correctly in either mode.

Your Time Capsule WiFi network may be quicker and faster than your modem WiFi network. This makes your local network faster, but it will not make your connection to the internet any faster.