You can't reach the Internet from either of your two PC's because they are on a different network (192.168.1/24 - I assume) than the LAN interface of your router (on 192.168.0/24) offering visbility to the Internet (your "gateway").
In order to cross networks, you need a router or a device capable of routing.
You didn't specify which ProCurve switch model you are using, but I am going to assume it's an older/standard layer 2 switch which does not do routing (a "layer 3" switch).
You'll need to either:
- Get a different router (than the DG834G) with the ability to configure and place one LAN interface in each network. i.e. An interface in each of your two networks (192.168.0 and 192.168.1).
- Place your Netgear router into the VLAN and configure the router's LAN interface to use the 192.168.0/24 network.
- Reconfigure your VLAN to use the 192.168.1/24 network and place the Netgear LAN interface onto your VLAN.
An analogy of what you're facing. Your PC's are one road - Call it Main Street. Your router is on 1st Avenue. To get to the freeway and the rest of the world, you need to drive your car from Main Street to 1st Avenue and then onto the freeway (information superhighway!).
To complicate things, cars on Main Street only have maps of Main Street. Similarly, cars on 1st Ave only have maps off 1st Avenue. If you went from Main Street to 1st Ave, you'd get lost cause you don't have a map.
So to drive your car Main to 1st, you'll need to get a new map as you cross from one road to another. In very generic terms, that's the job of the router. It tells cars how to get from place to another.
Your PC's don't know how to get to the 192.168.0/24 network and won't without something - a router - telling them how to get there.