If it's actually taking a while to kick in, no, definitely not normal. Pretty much by the time the Finder loads, the network connection should have kicked in (usually well before).
However, the fact that it eventually works points to it being something else--if it were actually faulty hardware, I'd expect it to not work at all.
For example, some types of broadband modems disconnect themselves from the service after a period of inactivity, and don't reconnect until there's an attempt to access the internet, at which point it takes a little while. If yours does this, perhaps it's not beginning its connection attempt until you try to get at the internet, and it's a little while before it actually kicks in after that?
One thing to try would be to immediately open Network Utility (in the Utilities folder) and see what it says about network connection en0. If the link status says that it's connected, then it's not the iMac that's delaying your being able to get to the internet, it's something the modem is doing.