I'd say get everything separate, apart from the NIC possibly.
Onboard video is almost always complete rubbish, onboard sound can seem OK but can actually cut your performance by up to 20% in some cases.
If you're into any modern games, get a separate video card definitely. Also grab a separate sound card if you're a hardcore gamer, but I manage fine with the onboard.
Like I say, you'll probably find the onboard LAN controller is fine. But if the LAN connector breaks, it's attached to your motherboard, so part of your board is broken. You might want to get a separate PCI NIC just for peace of mind, but I wouldn't say it makes any great world-changing difference.
Summary: Get a separate video card, and you're fine