Argue that the composition of two rigid motions is a rigid motion.
So $ T_{1} \circ R_1 \circ T_2 \circ R_2= T_3 \circ R_3 \space $ for some $ T_3 , R_3 $.
Determine formulas for $T_3$ and $R_3$
The idea makes sense but i'm not even sure how to start this argument.
Personally I'd start observing that a rigid motion is defined as a transformation which preserves lengths (and orientations), so if each step preserves these, then so does the whole transformation.
If you actually want to combine multiple transformations explicitely, I'd do so using homogeneous coordinates. Write your translations and rotations like this:
$$ T_i=\begin{pmatrix} 1&0&x_i\\ 0&1&y_i\\ 0&0&1 \end{pmatrix} \qquad R_i=\begin{pmatrix} \cos\varphi_i & -\sin\varphi_i & 0 \\ \sin\varphi_i & \cos\varphi_i & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Multiplying such a matrix with a column vector $(x,y,1)^T$ will result in the corresponding result $(x',y'1)^T$. Performing multiple such transformations in sequence can be expressed by multiplying the corresponding transformation matrices. So you can combine the matrices on the sides of your equation, use some known formulas to turn products of trigonometric functions into trigonometric formulas of the sums of angles, and thus obtain the parameters of the right hand side from those on the left hand side.