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$1225$.
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.