(That picture in your avatar is Weil, right? You should start by reading Weil's Number Theory: an approach through history).

FLT(3) is the assertion that the curve $x^3+y^3=1$ has three rational points (including the point at infinity). The standard process of putting an elliptic curve in Weierstrass form shows that this curve is $y^2=x^3-432$ if I remember correctly. Now use descent on this elliptic curve (maybe an isogeny of degree 3) to show that the Mordell-Weil group is finite of order three (see, e.g. Silverman for the general theory). This may be Euler's proof, maybe it's discussed in Weil's book.

FLT(4) is weaker than the statement that the equation $x^4+y^4=z^2$ has only the obvious solutions. Again, this becomes the problem of finding the rational points on $y^2=x^4+1$ which is again an elliptic curve and Fermat's proof is a 2-descent showing that the Mordell-Weil group is finite of order four. I'm pretty sure this is in Weil's book.

I have no idea what conics have to do with any of this.