How do you get a Bitcoin Public Key from a Private Key
I'll try answering this again in a different way,
using small numbers to keep it readable.

convert the private key to binary representation, so decimal number 105, which is 0x69 in hex, becomes  01101001.

calculate this list of points, by repeatedly doubling the Generator point G:
1*G
2*G = G+G
4*G = 2*G + 2*G
8*G = 4*G + 4*G
16*G = 8*G + 8*G
32*G = 16*G + 16*G
64*G = 32*G + 32*G


write the bits of the private key next to this list like this:
privkey    pointlist
   1          1*G
   0          2*G
   0          4*G
   1          8*G
   0         16*G
   1         32*G
   1         64*G


now start adding only those points which hava a 1 written next to them.
    9*G = 1*G + 8*G
   41*G = (9+32)*G = 9*G + 32*G
  105*G = (41+64)*G = 41*G + 64*G


now you have calculated the public key for privatekey 105 by using only point doubling and point adding operations.


The actual value will be:
(0xf219ea5d6b54701c1c14de5b557eb42a8d13f3abbcd08affcc2a5e6b049b8d63,
 0x4cb95957e83d40b0f73af4544cccf6b1f4b08d3c07b27fb8d8c2962a400766d1)

The public key will then be in one of two formats:

full, this is a 65 byte number, starting with a byte 0x04
compressed, this is a shorter form, 33 bytes, starting with either 0x02 or 0x03.

The full public key is 0x04, followed by 32 bytes of the x-coordinate, followed by 32 bytes of the y-coordinate.
The compressed public key is 0x02 or 0x03 depending on whether the y-coordinate is an even or odd number,  followed by 32 bytes of the x-coordinate.
Then some more notes on your question:

Some older texts used to refer to scalar point multiplication, as exponentiation, that is why sometimes the private key is referred to the private exponent.

the parameters of the curve were chosen randomly once. meaning that the designers of the secp256k1 curve tried to unsuccessfully convey that there is no specific structure to this curve. Meaning that the NSA could or could not have put a mathematical backdoor in the curve parameters.

when using this curve and generating your public key, -you- have to choose your private key randomly, in a way that it is impossible for anyone to guess it.

the generateor G is a specific point on the elliptic curve, defined in the secp256k1 curve.