AC is a sinusoid at 60/50 Hz. In principle, adding a DC current is just an offset, so e.g. if you had 1 Amp of AC current it would look like a sine wave 1 Amp high (actually sqrt(2)=1.414 high using the usual convention but that's not important) and the wave would be centered at 0A so go from -1(.414) to 1(.414). If you had a DC current of -1A on top of that, it would just be offset by that amount, so would look like a sine wave with a minimum of -2.414 and a maximum of .414.
Tldr, DC is just like an offset to the voltage or current waveform which is itself a sine wave.
Electrical conductors are basically reservoirs of electrons that can move around freely. Think of a body of water like the ocean. There is a lot of movement happening all at the same time: the tides, the waves, rip currents etc.
A wire is an essential one dimensional conductor; the electrons can move only backwards or forwards. Think of a narrow channel of water like a canal or river. A canal has no current, while a river has a direct current: it always flows one way. But a tidal inlet has an alternating current. The net flow of water is zero, but there is still a constant movement of water, backwards and forwards.
At a tidal estuary both things are happening: there's a DC component caused by net egress of water and an AC component caused by the tide.
The analogy doesn't really work because bodies of water also have capacitance. Wires are more like pipes. But hopefully you get the idea. Another way to think of it is like shouting in the wind, if you know how sound works.
Look into Fourier transforms if you're interested in learning more.
Can power lines have multiple currents in them at once? What would that mean for when the AC phase is moving opposite the DC direction?