What you have are 12/2 cables (if you are positive they're not 14/2) The ground wire does not count as a conductor in classifying this cable.
There are many 'standard' sizes of junction boxes. You'll need to know the volume of yours.
For #12 conductors, you need 2.25 cu in of box volume for each conductor. (For #14, it's 2 cu in) You currently have 7 conductors (only one of the grounds count) and want to increase it to 9.

Assuming the junction box is big enough, what's the proper way to connect the wires? Just use a bigger wire nut?

Wire nut allowed combinations are printed on the side of the 100 box of Ideal wire nuts.

Or you can use this, courtesy of our fine friends at Ideal Industries

Here is a link that might be useful: Listed Wire Combinations

Or you could try the push-in connectors on page 15 of the catalog below.

Here is a link that might be useful: Ideal Push-In Connectors

"Or you could try the push-in connectors on page 15 of the catalog below."

Wait till you see the price per connective.