Does electricity flow on the surface of a wire or in the interior?

It depends on the frequency. DC electricity travels through the bulk cross section of the wire.

A changing electrical current (AC) experiences the skin-effect where the electricity flows more easily in the surface layers. The higher the frequency the thinner the surface layer that is usable in a wire. At normal household AC (50/60hz) the skin depth is about 8-10mm but at microwave frequencies the depth of the metal that the current flows in is about the same as a wavelength of visible light

edit: Interesting point from Navin - the individual strands have to be insulated from each other for the skin effect to apply to each individually. That is the reason for the widely separated pairs of wires in this question What are all the lines on a double circuit tower?

Stranded wire is used because it bends more easily, but it has essentially the same conductive properties.

Current flows throughout the entire wire. This is easily tested by measuring the resistance of round wires - the resistance will fall quadratically with the radius, indicating that it's the cross-sectional area that matters.

Amendment: this answer is only correct for direct current - see Beckett's below for AC. The changing magnetic fields introduce eddy currents which yield the skin effect, where current tends to be carried only within the "skin depth" of the wire, which is not proportional to the radius.

This is a bit unrelated to the original question, but it's worth mentioning that this can arise as a common misconception due to the fact that static electricity accumulates on the surface of a conductor. While this is true, it's correct that current tends to flow through the bulk of a conductor, and current density is measured in units of $\text{A}/\text{m}^2$.

Also, Martin's answer makes a good point, the skin effect is relevant for AC currents, but unless you're dealing with inch-thick wire, it won't really make a difference. At higher frequencies, stranded wire might help a little bit, but it would still be susceptible. There are special ways to strand wire (like the litz wire to mitigate/negate the effect, but that wouldn't be needed for mains electricity.