I'm trying to add a 20 amp circuit breaker into my existing main panel. The main panel has one vacant slot to put the breaker, but the neutral and ground bars have no more vacant slots. My question is this -- can I hook up the new neutral and ground wires to an existing slots? Meaning if the slots are already being used by another breaker, can I also hook up my new neutral and ground wires into these slots since there are no more available slots? I also see that the neutral and ground wires are all mixed up on the two bars. I'm saying that neutral wires and ground wires are not hooked up seperately from the two bars. I guess they're okay if done in the main panel.
Yes, it's fine that the neutral and ground wires are intermixed. The ground and neutral bar and both bonded inside the service panel, so even if all of the ground wires go to the ground bar and all of the neutral wires go to the neutral bar, they are still electrically connected via the bonding jumper. It is only once they exit the service panel that they must remain separated for various reasons. For appearance reasons, however, it definitely looks tidier if the neutral and ground wires go to their designated bars.
To answer your main question, no, it is not usually acceptable to have two wires go to the same screw on the neutral or ground bus bar. The only exception to this case is if the bus bar is UL-listed for that purpose. Call the manufacturer if you want to find out, or you can just play it safe and use wire nuts to combine the two wires into one wire, and then connect that single wire to the bus bar.
So if more than one conductor can be placed under a lug, it must say so somewhere on the panel (or in the panel documentation).
If you have a bus bar on each side of the panel like in the diagram above, you should not connect the neutral to one bus and the ground to the other. Neutral and ground from all circuits on the left should go to the left bus bar, and Neutral and ground conductors from the right should go to the right bus bar.
So if the panel allows multiple conductors under a lug, you can terminate multiple equipment ground conductors under a single lug. However, you cannot use a single lug for multiple grounded conductors (neutrals), or a mix of equipment ground and grounded (neutral) conductors.
Here's why you can't have multiple neutrals in a single terminal.
And this is why you can't have a neutral and a ground in a single terminal.
You can , however, have both grounded conductors (neutrals) and equipment ground conductors connected to the same bus bar in the main service panel if the grounded conductor bar/bus (neutral bar) is bonded to the equipment ground bar/bus (it's different if you are dealing with a sub-panel, since the bar/bus will not be bonded).
Long story short
You'll have to check the panel documentation to determine if multiple conductors can terminate under a single lug. If they can. You can connect a couple equipment ground conductors to a common terminal, which should free up enough space to add the breaker.
NOTE: This only applies to bus bar terminals, most breakers are not rated to be "double tapped". So you should never have two conductors under a single breaker lug.