I think you’re confusing terms. You likely have a 2-in-1 modem+router from Verizon. You can probably keep that and use it for your modem (or ask them for an upgrade/replacement of modem-only hardware).
Your wifi router is the device that the Chromecast is connecting to and probably can’t because if it’s old it’s using an out-dated 802.11 spec. I believe the Chromecast wants at least 802.11n since it has to stream video.
As was mentioned by @Wheatums, the Apple Airport Extreme (or Time Capsule if you want network storage) is really nice (I have a Time Capsule and love it). The best part about it is it supports 802.11ac which will transmit both 2.5 and 5hrz simultaneously for capability without sacrifice of speed.
You don’t want the wifi to be your bottleneck, you want your connection out of the house to be.