If you're planning to use Ray's Arithmetic to teach math, sit down and read that teacher guide. You'll see more clearly how the books are meant to work together. In fact, you'll see that you don't even need to use a book for quite awhile, since you'll be working with counters to develop number sense.
When I start a young child learning math formally, we start with counters, just doing short lessons. I set out some objects and ask how many that is. I try to keep the number small enough that the child can tell me how many without counting. Once we have the number established, then I take one away and set it to the side. "Now how many are there?" We repeat this process for all the various combinations that make up that number. And the lesson is done.
When the child is pretty comfortable with 1-10 this way I often will start adding counters sometimes instead of just subtracting them. I think Ray's actually recommends working on number sense up to 20, which is probably much better. I just get impatient.
At this point, I may actually get out the Primary Arithmetic book and start asking questions from the addition or subtraction section, just using the word problems. At first, we'll do problems only for one number family and only in order, not mixed up. Once that's pretty comfortable or if the child notices the pattern, I will ask the questions out of order, but still staying on one page. If we need more practice, I will use a deck of cards with only the number cards. As I turn a card face up, the child has to add that number to the number we're working with that day. If the answer is wrong or slow in coming, the card goes back into the deck so we can encounter it again. This provides lots of quick practice and is much easier to use than traditional flashcards.
It's important to realize that this stage of math learning may take a long time. This is foundational and should be solid before moving on. There's no rush!
Once basic addition and subtraction are solid, you can progress to the more complicated exercises in the book, using your judgment about what to introduce. Refer to the teacher guide for guidance too.
Don't move on to multiplication and division until you see that addition and subtraction are well understood. At that point, I go back to the counters and start slowing building a multiplication table, on paper, with the child. Once we have a number family added to the multiplication table, we will do the word problems for that family from Primary, using the table as a reference. We'll keep practicing one number family at a time until the child is comfortable with that. If we need more practice, I will make up word problems. Peggy Kaye's Games for Math can be helpful at this stage too. Eventually we'll use the playing cards as flashcards in a similar manner to the way we used them with addition and subtraction.
Slow and steady is key. Don't move on until you see that the understanding is solid. When basic addition and subtraction are firm, you can do the more advanced addition and subtraction work in the Primary book, but you don't have to do every bit of it. When basic multiplication and division are firm, you can do the more advanced multiplication and division work in the Primary book, but you don't have to do every bit of it. You decide what is needed.