Sadly I took AP Physics algebra-based last year, when i am going to need a calculus based physics class since i want to go into an engineering field

So, I've been doing some physics on my own lately. I have ran into a weird problem.

Fg = (Gm₁m₂)/r²
Force of Gravity = (Gravity Constant ⋅ Mass of object 1 ⋅ Mass of object 2) / radius between the center of mass of two objects²

Of course the Force of gravity on an object includes the mass of such object, so we'll cancel m₂ on each side, leaving us with:

g = (Gm₁)/r²

For the sake of...me:
m₁ = mass of earth

If you graphed this equation, gravity relative to position away from the C.O.M. of m₁, you'd get:
(6.67E-11 ⋅ 5.972E24) / (X + 6.371E6)²

I added in the 6.371E6 to automatically include the radius of the earth.

I would like to figure out how fast a object is going at a certain time and what position the object is at, starting with no initial velocity, and any starting position.
My problem is trying to figure that out. I thought integrating (Calculus) would do the trick, but it gives me negative values that the limit to positive infinity is zero.
I also tried taking the y values (gravity at a certain position) and integrating those, but that does the same thing. The graphs do look different though.

Anyone know how to solve this?

Well, F = ma
And F is the sum of all forces, which in this case is just the force of gravity (Fg)
The force of gravity is equal to m*g, where m is the object's mass, and g is the acceleration due to gravity.
mg = ma
g = a
Then, you need to use vectors, which store the x and y components of an objects velocity. I use lists for this.
In my gravity experiments, I use the arbitrary designation of one frame updating being one second.
Finally, you will need to use trigonometry to determine the angle between both objects, and then use sines and cosines to get the x and y components of acceleration. Once you solved this, you can add your new acceleration to their respective x and y velocities.