There are many ways to do problems in mathematics so if you see someone doing it one way, just remember that it is not the only way to do a problem.
The way that you did it (the idea as well in general) as a good way. The number of permutations is as you said (720) in total is right.
Now for the number of permutations for no boys getting a gift and one boy getting a gift. We find those and subtract those to get the number of ways for at least two boys getting a gift.
So no boys getting a gift is 6*5*4 = 120, since you only have 6 girls and one boy getting a gift means you can have the boy get a gift either at the first step, the second step, or the third step. So calculate these individually we get:
1st step = 4*6*5 = 120
2nd step = 6*4*5 = 120
3rd step = 6*5*4 = 120
So total number of times 1 boy gets a gift = 360 and number of times boy gets no gift = 120 so these add to 480.
Subtracting 480 from 720 gives us 240 different choices.
So in this problem, we had no boy getting a gift and we also had a boy getting a gift at one of the three stages of getting a gift (and considered all three stages).
This technique also relates to what are called probability trees and they are useful for doing these probabilities. For this problem you have two branches: one where no boy gets a gift and one where one boy gets a gift.
Now the first one with no boy is a simple calculation with all the girls, but the one with one boy is different because each stage can be given to a boy (and the rest must be girls) so we have to look at the three cases individually, so we create a situation where we look at the three things individually and add them up.
In probability if things have nothing in common, then the number of things is added and all three cases are completely separate from one another.
The multiplication rule in probability should be explained so I'll do a short explanation.
Everytime you do a multiplication, the next thing in the multiplication is independent to the things before it. So in your example I had 10*9*8 and with regards to independence I had 10 choices first, then 9 but the 9 choices are completely independent from the 10 I had before and the 8 are independent from the one before that.
So when I start I have 10 choices and then 9 but they are independent choices. If you see rules like probability independence and multiplication, it's important to keep this in the back of your mind.
So as a suggestion, remember to break things up into their smallest parts and then add up all the parts together later on. If you do this, you won't have to rely on rules that are a bit fuzzy and you will also be able to understand your answer in a deeper way.