Between. Although connectivity to the moon should be fine, pings in the order of 3-4 seconds happen regularly on Earth, too. From personal experience I can attest that WoW is perfectly playable with pings of up to 5 seconds.
Network connectivity between Earth and Mars is going to be a slow affair, though, and will require relay satellites for when the Sun is between the planets.
A worst case round trip time of 45 minutes definitely won't be fun with TCP.
I think people shouldn't get their panties in a bunch. Letters used to take months to travel from one end of a continent to another for some 3600 years and people coped just fine. We won't have near-real-time chat, but the rest of the internet ecosystem will work as before. Mail, forums, cat videos, etc. pp. will work fine.
The engineering problem is already solved for probe communications anyway.
UUCP was designed for networks where many nodes were only online for a couple of hours each night and then over slow analog lines. Perhaps it has lessons to teach us.
Bandwidth to/from Mars wouldn't be too bad with enough power output. Latency is the big problem.
So I think there will be massive caches on both sides. Accessing any popular content will be pretty fast because it will already be cached. Live data will be delayed by up to half an hour depending on orbital position, but that's not terrible. Conversations would be the worst, with up to an hour delay round-trip.
Past Mars it would get harder. You'd need much more power to keep the bandwidth up, and the latency would get worse.
