The best way to learn a singing game, in my opinion, is the most authentic: simply watch others playing it and then join in. However, when teaching in an elementary classroom, you will need to teach all of the students the singing game "from scratch" (although you could have the students watch a video of the game). So, there are a few approaches that seem to work, depending on the singing game. One approach is to first get the students into the proper formation (lines or circle) and then act as the one-who-is-it. For snowball-type games such as Rig a Jig Jig this can be accomplished with little or no instruction. The students will catch on quickly. After a few turns, someone else can start taking on the role of the one-who-is-it. Another approach is to teach the entire song by rote before playing the game. To teach a song by rote, sing the song for the students at least 10-12 times giving them something to do (see 100 Ways to Repeat a Song) or listen for each time they hear the song. The things to do should be connected to each other from one song to the next. For example, if they are adding actions, they should probably continue adding more actions throughout or something related, rather than all of a sudden switching to identifying specific rhythms. I outlined such a sequence in the post for Four White Horses. Then, after they know the song, on a different day usually, teach them how to play the game with as little talking as possible. You will discover many things that work. The important thing is to let the students make their own sense out of complexity as much as possible, give verbal instructions as little as possible, and keep it interesting by giving the students something meaningful and engagingly challenging to do throughout the process. Then, on subsequent days, play the game again. Eventually you may want to change the game or have the students change the game (see Four White Horses and Drunken Sailor, for example). You can also explore the song in more detail in order to address specific curriculum standards. For exploration ideas, see 100 Ways to Repeat a Song.
This is such a fun way to teach. Remember, though, that what works for me might not work for you and your students. Be creative and try new things. Enjoy!