how to find the distribution density of the variable X + Y? X and Y are independent and have the same exponential distribution wih density: $$\lambda e^{-\lambda x} 1_{(0,\infty)} (x) $$ The density of the X + Y distribution can be calculated by convolution of the functions: $$f*f(t)=\int_{-\infty}^{\infty} f(x) f(t-x) \, dx $$ but I still have no idea how to solve it, during the lecture we only gave an example for a uniform distribution on (0,1)

Applying the convolution formula to your example you immediately get
$$f_Z(z)=\int_0^{z}\lambda e^{-\lambda x}\lambda e^{-\lambda (z-x)}dx=\lambda^2 z e^{-\lambda z}$$
$z\geq 0$
Proof:
By Definition you get
$$F_Z(z)=\int_{-\infty}^{\infty}f_X(x)\Bigg[\underbrace{\int_{-\infty}^{z-x}f_Y(y)dy}_{F_Y(z-x)}\Bigg]dx$$
Derivating you get the density
$$f_Z(z)=\frac{d}{dz}F_Z(z)=\int_{-\infty}^{\infty}f_X(x)f_Y(z-x)dx$$
To realize that the integral bounds are $[0;z]$ just do a drawing of the transformation function


Without any calculations, $Z$ density could be found using the properties of Gamma distribution. As $Exp(\lambda)=Gamma(1;\lambda)$, thus $Z=X+Y\sim Gamma(2;\lambda)$
