There are different ways one can go about using sign language in Primary. I'll give a few of my ideas and then you can take the ideas and run with them!
First of all, if you're using sign language just for learning you don't have to worry if you don't want to about which hand the kids use and they will just do it with the hand that they want. However, if you are wanting them to learn with their dominant hand (for most it will be the right hand), or all of them to do the signs with the same hand, you can tell them to mirror you while you do the signs with your left hand (backwards, unless you are left-handed). To help them practice this I first do a silly mirroring activity by telling them to mirror what I do and then I move my arms in silly ways (such as touching my nose with my elbow sticking out, patting my head, raising either arm, etc.). This would be good to do if you plan on using sign in a song for a presentation in Sacrament meeting, etc.
Sometimes I will start with the more familiar part of the song to teach, such as the chorus. I usually do only a few signs at a time, sometimes that's a phrase and sometimes it's half of a phrase. If it's a song they are not familiar with I will sing the phrase to them first and have them repeat after me. Then I will show them the signs for the phrase and kind of explain how to do each one and then have them do it with me, while speaking the word it goes with each time. Then we will sing and sign the phrase we've learned, adding phrases together and singing from the beginning to where we've learned until we have learned the whole thing and can sing it all the way through. If it's an easy song they know another alternative is you can have them follow along with you as you sign each phrase and then slowly go through the signs after they've seen them.
While I'm teaching the signs we sing small phrases a cappella, or the pianist will play just the melody of the song. When we put phrases together the pianist usually chimes in with either the melody or accompaniment. It's nice for the kids to have the music in the background to stabilize their learning of the melody of the song.
WHEN TO TEACH THE SIGNS
It depends on the difficulty/familiarity of the song when I begin sign. For the hymn I Stand All Amazed, for instance, I began teaching only a few repetitive signs the first Singing Time (I did this at the end of another lesson plan by Pat Graham), and will add the rest in the next Singing Time. I taught He Sent His Son (which is more well known by most kids) this year with a different activity the first week and then went over it again with them and tuned it up the second week with some sign language. I've also taught a song with sign language in one Singing Time.
Sign is a great way to teach a song on the first week, and it can also be a great way to review or refine a song on the second week. Personally I would either do the first or second week of teaching the particular song and not wait longer. The kids will be happy to do the signs in later weeks as you continue to sing the song, and the more they review it the more they will learn it and get it down.