@Brenkj I was in the same boat as you when I first got my model 3. At first I tried the regular outlet and like what other people here have said before, it charges slow. If you commute 10 miles per day, this can always be an option where you charge daily. However, keep in mind that you will be tied to this schedule. Having said that I knew I had to go with a faster charging mechanism i.e. nema 14-50 or wall charger.
I got a quote from 3 electricians to see how much to add another outlet, nema 14-50 in this case, and the quotes ranged from $800 to $1600. The $1600 quote was adding a sub-panel with the nema 14-50 outlet dedicated to it.
Some information regarding your panel. The main circuit breaker, that has the 200 label, is the amount of amps that panel can support. You have to follow the 80-20 rule when talking about the circuit breakers. This means that if a breaker is labeled for 20 amps, you can draw 16 amps out of it. That's why if you have the nema 14-50 on a 50 amp breaker, you can draw 45 amps continuous, which will charge your car faster. Just apply the same concept to the wall charger connected to a 60 amp breaker or 100 amps. In my opinion, the nema 14-50 connected to a 50 amp breaker would suffice most of the Tesla owners here. Keep in mind you can always use other chargers as a backup plan.
Your panel is already full but you can combine some circuit breakers together and use a tandem. Just search for "tandem circuit breaker" to know more about it. This is when a professional electrician will be worth hiring/consulting so they can properly determine which breakers to combine. Keep in mind that your panel supports 240V, occupying one slot makes the breaker use 110/120V. That's why you'll see your oven or hvac occupying 2 slots since they run on 220/240V. Just a hint, if you have to combine single breakers into tandem, you need to pick alternating breakers so that they are on the same line.
Regarding the length distance of the outlet to the breaker. The closer it is to the breaker, the better. Those 6 awg wires cost a pretty penny. In addition if you can make it close to the breaker, you have less wire being exposed. Bear in mind that these wires are high amperage and best to make it short as possible.
To minimize costs, you can buy the stuff yourself and have an electrician wire it in. But if these stuff still go way over your head, best to have an electrician do it for you. IMO and to be on the safe side, just hire a professional who can make sure that the load will be to code.
Good luck and enjoy your Tesla.