Two ways to get four conductors twisted tight.
First, strip a lot more than think you need, I typically end up with roughly a 1"+ of bare copper, on a solid #12. Now you take two pairs of wires, and hold them so that they form an X. They need to be less than 90* apart, closer to 45 to 60* is better. The wires need to cross tight to where the insulation starts. Hold this in your non-dominant hand. Now take the lineman's in the other hand and get a good bite on the bare conductors. The tip of the pliers needs to cover where the wires cross, almost touching the insulation. Now grip the wires tight, and pull the lineman's toward you, while twisting clockwise. Make a half turn, release the pliers, get a new grip and repeat. Do this until you have a nice full twist to the end. Now cut the end off, leaving about 5/8" +/-. Next take the lineman's and twist the end of the cut. This rounds the sharp, squared off, cut end and gives the group a bit of a "thread" to help the wire nut get started.
The second way is to "add" a wire, if you can successfully twist two or three conductors, you can leave the group slightly loose, and add another wire to the bundle. You cross the new wire so that it's parallel to the threaded group and the insulation is even, then you draw into the group by pulling and twisting it tight with the lineman's.
After a bit of practice with the first technique, you will be able to easily tackle 5-#12s as long as they are all solid wire and the same gauge. When it comes to mixing wire gauges, it's a lot tougher, and often easier to twist the same size together and add the odd one in. Solids and stranded together can really suck. If it's a single solid to a single stranded, pre-twist the stranded wire with the lineman's and twist it hard. This keeps stray stands from driving you batty while you are installing the nut.Then hold the two wires so that the stranded is about 1/8" ahead of the solid before it enters the wirenut. There is little to be gained by attempting to pre-twist this combo. together before installing the nut. If it's several solids and a stranded. Twist and prep the solids, pre-twist the stranded and then enter them into a wire nut.
As for gouging or damaging the wires, it generally isn't an issue, unless you are doing as budgetslasher describes. This is called "Ringing" and it doesn't matter if you do it by misusing a stripping tool, or skinning wires with a knife, it is a real big deal. On larger conductors, like the ones feeding the main breaker in a service, ringing can cause hot spots and fires. On smaller conductors, it can weaken them until they break at the score.
I hope this helps. It's fascinating to discover how hard it is to describe something that you can do with you hands, and not give a second thought to, then realise that putting all that in words, that you hope makes sense, is another animal entirely. Good luck.