Text messages, yes, because SMS is not encrypted. iMessages, and FaceTime, no because they are end-to-end encrypted. Likewise, most other messaging apps: Signal, Telegram, WhatsApp, Skype, Google Meet, etc.
Internet activity, sort of; they know your IP address and the IP address you are accessing, but not the content exchanged during that access. You can further limit what they know about your IP addresses using VPN, but then your VPN provider knows that information, and Verizon is probably more secure than many VPN providers.