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Connecting Vlan to Internet Access

My goal:

A network divided into E.g. two Vlans that both have internet access but cannot talk to each other.

My problem:

First try:
Internet -> Fritzbox -> Managed Switch -> Clients
Didn’t work. None except the default Vlan had Internet access.

So I thought, maybe its because the router doesn’t understand Vlan tags.

Second try:
Internet -> Fritzbox -> TP-link router firewall -> Managed Switch -> Clients

Sadly I cant configure the Fritzbox in Brige mode, to only use it as a modem. The TP-Link router is a router behind a router, and I’m not sure if that’s part of the problem. But even now with a switch and configured Vlans on both of them I cant get internet access on any Vlan except the default one (1). Yes the Trunk ports and VIDS are correctly set.

I am probably missing something very obvious but I cant get a hold of it. How do I fix this ?