Hello,
the link(s) between the switch and the router need to be trunks only when you use subinterfaces on the router (called router-on-a-stick), which you don't have configured. You are using the physical interfaces on the router, so the ports connecting the switch to the router need to be access ports.
Here is the correct switch and router configuration. For the sake of convenience, I have configured two DHCP pools, so if you set the PCs to DHCP, they will automatically get an IP address:
Switch
interface FastEthernet0/1
description Link to PC1
switchport access vlan 10
switchport mode access
spanning-tree portfast
!
interface FastEthernet0/2
description Link to PC2
switchport access vlan 10
switchport mode access
spanning-tree portfast
!
interface FastEthernet0/3
description Link to PC3
switchport access vlan 20
switchport mode access
spanning-tree portfast
!
interface FastEthernet0/4
description Link to PC4
switchport access vlan 20
switchport mode access
spanning-tree portfast
!
interface FastEthernet0/5
switchport access vlan 10
switchport mode access
spanning-tree portfast
!
interface FastEthernet0/6
switchport access vlan 20
switchport mode access
spanning-tree portfast
Router
ip dhcp excluded-address 192.168.10.1
ip dhcp excluded-address 192.168.20.1
!
ip dhcp pool VLAN10
network 192.168.10.0 255.255.255.0
default-router 192.168.10.1
ip dhcp pool VLAN20
network 192.168.20.0 255.255.255.0
default-router 192.168.20.1
!
interface GigabitEthernet0/0
ip address 192.168.10.1 255.255.255.0
duplex auto
speed auto
!
interface GigabitEthernet0/1
ip address 192.168.20.1 255.255.255.0
duplex auto
speed auto
