Basic connection of "solid" (typically 14 gauge) to solid wire (by solid I mean single core) : strip a good inch of each. Lay metal bits side by side, grab the tips with pliers and twist clockwise (same direction you will screw on the wire nut). The twisted wires should appear to be like a threaded screw in the direction the nut will tighten on. If after twisting the ends are not neat, don't try to crimp with pliers, instead snip off the forked bit down to a neat nub with wire cutter. Screw on wire nut to hand tight. Wrap electrical tape clockwise (tightening the wire nut, not loosening) around the nut and then around the wires.
Connecting "stranded" to stranded (stranded = many small cores making up the wire). Strip about an inch or slightly less. Lay both wires side by side and fan them out flat (undo any twisted). Mush they fanned out wires together to make like one big stranded wire. It may help to tape the insulated part together just to hold the wires in place as you do this. With your fingers gently twist the big stranded wire clockwise. This is just so it forms a neat tip, not to create threads. Screw on wire nut & tape.
Connecting stranded to solid : this is by far the weakest of the three connections, and ideally you would use solder and/or crimp connection or perhaps a screw terminal block. Another option is to solder the end of the stranded wire to make it effectively a solid end, then wire nut it. But assuming you don't want to do any of that, you do this : strip stranded wire to 1.25 inches, solid to 1 inch. Twist stranded wire by itself to give it a neat solid end. Lay the two wires side by side with the stranded extended slightly past the solid, 1/8" or so. Tape the insulation of the two wires together just to hold them together. Do not twist the wires with each other. Screw on wire nut, then wrap in tape.
In all cases you can test the connection by giving a little tug, there should be no feeling of looseness (and if the little tug was enough to wreck it, it was no good). Wire nut connections only work great between wires of roughly the same gauge.