Amen to this.

I am a "homeschooling" parent. This does NOT mean my children are taught solely by myself and/or my wife, and it does NOT mean they are taught solely at home. It DOES mean that we have personally selected and combined a number of different educational opportunties for them. These include (but are not limited to):

Enrolling in college coursework while still in high school. Example: Harvard Math 23b. The majority of students in this class are admitted Harvard freshmen, but it is also available in an open enrollment capacity through Extension for anyone of any age willing to pay tuition. I like that peer group for "socialization" a whole lot better than the kids at my local public high school.

Hiring the chair of the language department at a local private high school to come to our home to provide personalized one-on-one instruction in classical Greek and Latin.

Hiring multiple music teachers for piano, guitar, theory, and composition.

Participation in team sports at the local health club.

Engaging a flight instructor for our son to earn a private pilot's rating.

Successfully completing qualifying flights for TARC

The Internet (Obviously). Taking advantage of online educational programs such as AOPS and edX and Open Courseware

Stocking our home with thousands of quality print books and plenty of subscriptions to lots of quality print journals (e.g. Economist, Nature, Lapham's Quarterly, IEEE publications, etc.)

Buying a whole bunch of the Great Courses

Joining CTY

Plenty of socratic dialogue with Mom & Dad. And plenty of unstructured time.

Flexibility to travel (including abroad) during the school year.

Concrete advice for OP: First, read The Underground History of American Education. Make of it what you will --- just include it (or criticisms of it) as a data point. Next, decide if any your local school choices (either public or private) are awesome. Do they approach the quality of Exeter or Boston Latin or Bronx Science? Understand the concept of a feeder school and that this concept can start at the elementary level. Got great public or private school options you like and can afford? Go for it. Not so much? Then go ahead and homeschool kindergarten. I guarantee you that your drop-out wife is capable of teaching your child to read and anything else they are supposed to learn in kindergarten. I guarantee you that unless you are completely negligent that your child will (if you choose) be able to enter first grade after a year of homeschooling and do fine. And I guarantee you that after a year you will be in a much better position to understand if more homeschooling is the right choice.

In theory, learning math independently (as opposed to taking courses or hiring a tutor) basically boils down to (1) obtaining some good, relevant books, and (2) actually doing enough problems in said books to learn the material.

With that said, the quality of available books varies widely. Some are much better suited to independent study than others. Some books simply focus on showing you suitable algorithms that will equip you well enough to "solve" routine problems, while others focus more on providing a theoretical basis for the material and making sure that you actually understand what's going on. Books of the latter sort are typically more work, but with a higher payoff.

Here are my specific book recommendations for learning high school mathematics and Calculus. The bias is toward being thorough, covering all the theoretical foundations, and assuming that you are willing to do a lot of hard work (though with very high payoff!). If your bias is toward just memorizing a few key formulas or getting off easy, this is not the right list.

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) by George Polya
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa translated and adapted by Will Harte
Algebra by I.M. Gelfand, Alexander Shen
The Method of Coordinates by I.M. Gelfand, E.G. Glagoleva, A.A. Kirilov
Functions and Graphs by I. M. Gelfand, E. G. Glagoleva, A. A. Kirillov
Trigonometry by I.M. Gelfand, Mark Saul
Basic Mathematics by Serge Lang
Kiselev's Geometry / Book I. Planimetry by A. P. Kiselev (Author), Adapted from Russian by Alexander Givental (Editor)
Euclidean Geometry: A first course by Mark Solomonovich
Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra by Tom M. Apostol
Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications by Tom M. Apostol