1. Always check with the arrival GA. Your connecting flight may take a last second delay and there may still be a seat available. It may be better to sprint than to argue. Do understand that the major carriers no longer call the other gate. There is no need. The departure gate agent can see that you have arrived and operations will make a decision about holding the onward flight (very, very unlikely) and isn't going to base that on hysterical phone calls. When seconds count, wasting them on "call the gate" stuff is silly.
2. Depending on the carrier, whether it is international or domestic and how complex the routing is: do not get yourself directly involved. The worst screw-ups occur when one agent is reissuing a ticket (which may be necessary) while you are on the phone making other changes. It is pretty routine to have the passenger wind up with a new reservation, but the ticket attached to the old reservation and then unable to board. Make a choice. One or the other of you deals with the carrier.
3. Always have alternatives in mind. Start with the carrier, then with the alliance and then other carriers. You are much more likely to get something palatable if you can feed the agent exactly what you want.
4. Get on the phone and get in line. Whoever helps you first, stick with them. It's not either or. All of this suggests having backup battery power for your mobile.
5. As others note, don't hang around the jetway for luggage. Find an agent and then worry about retrieving the gate check.
6. If in a real mess, evaluate (presuming you aren't a member), the value of paying for one-time lounge access (presuming domestic location with a lounge). Spending $50 or thereabouts may save a whole lot more.
Keep the stress out of the conversation. No matter how desperate, losing your temper, telling agents how important you are or other things not directly relevant to the immediate problems) gets you lousy service. Even if the entire incident is 100% the carrier's fault and was easily avoidable, pick your battles. If you want to complain when you get home, that's fine. But, in the moment, if your flight has left, it's not coming back and the agent can't build you a new plane on the spot.
7. If you are stuck overnight and the misconnect is for reasons beyond the carrier's control, e.g. weather, you are likely on your own in the US. Ask for a local phone # for distressed traveler hotel rates and you may find a decent deal somewhere nearby with a free shuttle. If you can't take the risk of eating the cost of an occasional overnight, make sure that you have travel interruption insurance through one or another source.