Is there a tool especially designed for opening a tube of caulk, silicone, or construction adhesive?
A knife to cut the tip at the desired spot, and a nail to puncture the seal inside. You'll use the nail later to hold the partially-used tube.
I remember as a kid never having any of that built into the caulking gun. Later, when I tried one that had a handy "cigar clipper" thing on the handle, I thought it was terrible, not allowing easy selection of the place on the cone and impossible to cut at an angle.
In a pinch you use the cutter on the side of the caulking gun, but a utility knife should be used if you plan to do detailed work vs just slopping the caulk on. (The puncturing wire on the gun is fine, though, if there is a seal in the cartridge.)
To reseal the tube, take a short piece of duct tape and fold it over the tip, with the fold over the end of the tip, and press the tape to itself on both sides. Don't fold tape tightly against the tip, but leave maybe 3/8" "headroom". Then, once the tape is in place, pump the gun until the tape balloons out slightly around the tip. (Or you can now buy a tube sealer gizmo that looks pretty neat.)
With your teeth! It's a little known fact that per consumer ease mandates, all products sold in a cylindrical tube longer than 6 inches with a graduated nozzle must have a "yield" extending down at least 5/8" along the nozzle structure. The yield, by regulation, must tear cleanly when penetrated by a dull edge at less than 120psi. The statute was put into effect after it was revealed that certain containers housing solvents used in medicine could not be opened in the case of an emergency. The specific technique, referred to as "occlusive puncture" employs a combination of the lateral incisor and upper and lower cuspids, and is taught primarily to medical professionals and military personnelle. If performed properly, it will even be effective on potato chip bags and DVD packaging.