Purism doesn’t provide internet service or cellular network service, but you should be able to use your current providers.
Purism can’t test all the cellular networks and you have to provide the SIM card from your cellular network provider. That being said, tests of the development phone models (Birch and Chestnut), found that most of the cellular networks are compatible with the Librem 5 (using the BroadMobi BM818 modem). Once the Librem 5 (Evergreen) starts shipping, people on the forums will give more feedback on what cellular providers work with the Librem 5. See: Will my existing SIM card work? What countries and networks will be supported? – Purism
You should be able to use your standard internet through the WiFi on the Librem 5.
Purism plans to offer the PureOS Store which will only contain mobile apps which will work in Phosh (the Librem 5’s mobile interface). However, the mobile app selection will be very limited when the Librem 5 is first released and it will take Purism and the community time to add more apps. If you have particular types of apps that you will need on your phone, you should ask about those.
You can also install most of the software in the Librem 5 that is found in the Debian Linux distribution, which contains 57,703 packages, but most of that software is designed for a Linux desktop, so it won’t be easy to use in a mobile interface. It will take time to adapt the GTK desktop apps so they can be used in a mobile phone interface.
If you want to check out what apps are currently available, then download the Librem 5 image and run it in the Qemu virtual machine on your PC.
The Phosh mobile interface on the Librem 5 is very easy to use in my opinion. If you can use an Android or iOS phone, then you should have no trouble using Phosh. However, you need to expect limited app selection and there will be many software issues that still need to be resolved. For example, Purism is working on a new camera app, but until it is implemented, expect the camera interface to be very basic.