This is an area where, as the speaker or writer, you have a lot of leeway to decide whether you view the paired subjects as fundamentally separate things or as things to be viewed in combination. For example, "going to the stadium" and "watching the game" are clearly two actions—one involving travel to the venue and one involving what one does after arriving—but they are closely allied as aspects of attending the sporting event, so it would hardly be surprising to hear someone emphasize the connection with a construction like this:
Does going to the stadium and watching the game appeal to you?
On the other hand, in situations where the connection between the actions is more remote, we might expect the separateness to be reflected in the verb used:
I don't know... Do running out of gas halfway there and sitting in a downpour at the stadium sound like fun to you?
The actions "surfing the internet" and "using social networking websites" fall somewhere between the smooth continuity of "going to the stadium and watching the game" and the far less closely bound association of "running out of gas halfway there" with "sitting in a downpour at the stadium." That being the case, I think that you are free to use either verb, according to your preference. Either way, your hearers or readers will have no trouble understanding what you mean; so the main effect of your choice of verb will be to emphasize either the connectedness of the two actions or the separateness of them.