The notion of considérablement is indeed the right translation for this sentence. As we can see in these translations from linguee, we focus on the “big effect” side of the action.

If I wanted to translate this sentence in a Frenchie way, I would say :

• Utilisez ce livre, et vous progresserez à pas de géants

This translation is more respectful of the literal side of the sentence.

Other ways to say it :

• Utilisez ce livre, et vous deviendrez maître en la matière

• Utilisez ce livre, et vous atteindrez des sommets

I think the closest thing to leaps and bounds for French would be : "À grands pas"

Utilisez ce livre et bientôt vous progresserez à grands pas

Literally means "with big steps" but it’s often used to point out how fast things can progress

Science is making leaps and bounds

La science progresse à grands pas

EDIT : Fun fact, "grand pas" was also the translated term used for Neil Armstrong’s famous first step quote on the moon

"Un petit pas pour l’homme, un grand pas pour l’humanité"

"One small step for [a] man, one giant leap for mankind"

So "grand pas" can also mean "a big/giant leap" in progress

You misread your reference: the idiom is not “leaps and bounds” but “by leaps and bounds”.

The idiom, “by leaps and bounds”, has been traditionally used as a complement for verbs : “to do something by leaps and bounds”; “to make leaps and bounds” is a distortion of the original expression that has come into being recently and it might not be found acceptable by all. It sounds very awkward to me.

A standard English formulation would be

• Use this step by step book and you’ll soon be progressing by leaps and bounds.

In French you do not say anything if you use a literal translation. “Vous ferez des sauts et des bonds.” has no figurative meaning. Nevertheless, the traditionnal expression has a literal translation and you can say

“faire des progrès/avancer/progresser/aller de l’avant/… par sauts et par bonds”. (this is to be found in the Harrap’s dictionary)

So, because of the proximity one might easily guess what you mean if you say “faire des sauts et des bonds”.

Your last sentence is then perfectly acceptable in French and as you’d wish to have it if you just change “considérablement” into the translation of the true idiom;

• Utilisez ce livre et bientôt vous progresserez par sauts et pas bonds.

The sentences shown here are reverso confirm this usage. This appears to be a recent acception, as there is another mentioned by the TLFi and that is the one found in the older litterature;

2. Au fig.
   a) Élan imprévu et isolé. Procéder par bonds, n’aller que par sauts et par bonds

To me (not an native English speaker), “making leaps and bounds” conveys the idea of “how fast things can improve (if you follow the steps we recommend in the book)”. It’s similar to:

1. “into high gear”
2. “at great speed / at the speed of light / break-neck speed”
3. “zooming ahead” (like in learning / teaching / researching is zooming ahead)

In addition to “vous progresserez à pas de géants“, or to replace it, I’d say/use “vous apprendrez / progresserez à vitesse grand V“. It means that you’ll learn (much) fast(er).