VLAN problem

You cannot get two different vlans to communicate with each other using only a switch. They are different lans, different networks entirely. And how do you get different networks to communicate with each other? By using a router. You need to configure a router to route between the two vlans.

To have 2 vlans talk to eachother you will need to "route" between them.

You need a router.

If you had a cisco router, You would plug that into the switch. Turn the link to the router into a vlan trunk and configure ip addresses of 2 vlan interfaces on that cisco router.

The router will then do inter vlan routing.

Yes Layer3 switch or a router is the only way for inter vlan communication. There is a very little difference between these two devices. The main difference is the hardware technology used to build Layer3 switches. It merges that of traditional switches and routers, replacing some of a router's software logic with hardware to offer better performance in some situations. So they are also faster than the routers. These layer3 switches are also cheaper as compared to routers but they do not have WAN ports and WAN features like routers.