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I'm redoing some of the outlets in our house, as they are pretty loose. Some of these have been really easy, but I came across one today where there were 3 hots and 3 neutrals. All 3 were going into the quick-connects in a outlet which looks like this:

Don't mind the destruction on the quick-connect in the picture, as I broke that taking the wires off and am not re-using this outlet. My new outlets look like:

For my other installations I've been using the screws on the side rather than the quick-connects, but since I had 3 hots/3 neutrals I figured I'd try to put 4 on the screws and the other 2 in the little quick connect holes. However, this plan doesn't work because it seems that the wire is too large for the quick connect holes on my new outlets (maybe a 14 gauge vs 12 gauge difference? )

What do I do in this situation? Do I nut two of the hots and 2 of the neutrals together? If so, can I determine the exact type of nut I need with my wire stripper (AWG of the bare wire correspond to the gauge of the wire or do I need to include insulation?).

Edit: A picture of the sides of the outlet I'm replacing, in which it looks like the tabs are present:

Your two options here are to pigtail the hots and neutrals, or get a new "backwire" outlet. A backwire outlet has small wire clamps on the sides of the outlet so the wire can be inserted straight in rather than looping around the screw. This type of outlet might be sold as "heavy duty", but they are great for handling up to 4 conductors on each side. The ground will still need to be pigtailed, but that's usually not a big deal.

Example outlet

If you look closely at that picture, you can see 4 holes in the back on each side. Wire inserts straight into the holes and the screws on the side tighten the wire in clamps. They are very similar to your old push-connect outlet, but they clamp tight on the wire.

Backwire outlets are great for smaller boxes because you don't need to worry about a couple 4-wire bundles of 12GA wire getting pushed into the box along with the outlet.

First, watch out for common gotcha's in changing outlets or switches. If any receptacle tabs have been broken off, this is a whole different ballgame.

I am assuming you didn't mess with any of the other wires that were pushed back into the box and nutted to each other. If you did, and especially if the receptacle has been split, one of the black/white pairs might be a switch loop, which would make a big mess. You'll find out; ask further if you have any trouble.

Now if you look at the ground wire on the receptacle, you notice it is pigtailed. That is, all the grounds from the cables join in a splice to a short wire that comes out to this receptacle.

The same thing can be done with hots and neutrals. Obtain some wire (buying 2 feet of #12 Romex usually gets you plenty) and attach a white wire to a silver screw, and a black wire to a brass screw. Then bring the outlet to the junction box and use 2 red wire-nuts to splice the black with the other blacks, and the white with the other whites.

See, what you're trying to do there is use the receptacle itself as a splice block to connect several other unrelated wires. That's not mandatory at all; you can use a wire nut instead. Now if you really want to use the receptacle as a splice block for up to 4 wires per side, look for "screw-to-clamp" type receptacles which permit that.

The best solution to this problem is to pigtail the outlet: wire-nut all three hots together with a new, short length of 12 gauge black wire, and then connect that to the outlet. Do the same with the neutrals, using a short length of white. Ground can be either green or bare.

As to which wire nut to use, that may vary a little based on brand, but usually it will be red. If you're using Ideal wire nuts, you can check the size you need on this PDF. For other brands, consult their instructions.