Q:

What number am I ? (square number and triangle number)

I'm a square number,
I'm also a triangle number,
You can write me as the sum of 2 square numbers,
You also can write me as the sum of 2 triangle numbers,
You also can write me as the sum of 1 square number and 1 triangle number,
I'm the smallest number of my kind.
What number am I ?
Bonus Puzzle : Find the second smallest number.

A:

You are

 $1225$.

This is a square triangular number, i.e. both a square number and a triangular number. It is also a sum of two squares, since

 $1225=5^27^2$ contains every $(4k+3)$-type prime factor an even number of times. (Explicitly, $1225=21^2+28^2$.)

And a sum of two triangular numbers, since

 $8\times1225+2=9802=2.13^2.29$ contains every $(4k+3)$-type prime factor an even number of times - see this Maths.SE post. (Explicitly, $1225=T_{19}+T_{45}$.)

And a sum of one square number and one triangular number:

 $1225 = 136+1089=T_{16}+33^2$.

It's the smallest square triangular number which is a sum of two squares, since $36$ isn't a sum of two squares.
Bonus round: the second smallest example is probably

 $41616$. This is the next smallest square triangular number; it's also a sum of two squares, since it's $2^43^217^2$; it's a sum of two triangular numbers since $8*41616+2=2.5.13^2.197$; I haven't verified whether it's the sum of a square number and a triangular number.

