When I connect to a client's VPN (Cisco), it takes a while for my browser to open. Are my browser requests and emails routing through the VPN connection, or is it just that my machine is having diffculties resolving Internet access with the VPN open?
Probably the first, all of your internet traffic is routing through your VPN. There are ways that you can set up routing to avoid this situation, but they are significantly complicated enough that most IT admins just use the VPN as your default route. There may be a way for you to configure your machine to only route VPN traffic to the VPN, but I don't have enough experience with VPNs to help you further.
Generally your web traffic does not go through the VPN. Admins normally set VPNs up for split-tunneling which sends company traffic across the VPN but all other traffic out your Internet connection, like normal. However, it is possible that all of your traffic is going across the VPN and through a proxy. That would explain why it's a little slow. Quick way to test is to right-click on the yellow-lock icon in your Task tray. Click on Statistics. Then click on the Route Details tab. In the Secured Routes column, if you see 0.0.0.0 0.0.0.0 then you have everything sent across the VPN. Don't want to get more technical than that. Hope that helps.
