I suggest two different ways of trying to solve this, which will give you experience of different bits of the formulation of Quantum Information Theory. I'll give examples that are closely related to the question you asked, but are not what you asked so that you still get the value of answering the question yourself.
Long-hand Method
Represent the kets as vectors, the Pauli matrices as matrices, explicitly perform the tensor products, and multiply everything out. So, we represent
$$
|0\rangle\equiv\left(\begin{array}{c} 1 \\ 0 \end{array}\right)\qquad|1\rangle\equiv\left(\begin{array}{c} 0 \\ 1 \end{array}\right)
$$
To calculate the tensor product, such as $|01\rangle$, we do
$$
|01\rangle=|0\rangle\otimes|1\rangle\equiv\left(\begin{array}{c}1\times\left(\begin{array}{c} 0 \\ 1 \end{array}\right)\\ 0\times \left(\begin{array}{c} 0 \\ 1 \end{array}\right) \end{array}\right)=\left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right)
$$
Remember that $\langle 01|$ is just the Hermitian conjugate of this,
$$
\langle 01|=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \end{array}\right)
$$
Then you do something similar for the operators. For example, $$X_1X_2=\sigma_1\otimes\sigma_1=\left(\begin{array}{cc} 0\times \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) & 1\times \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \\ 1\times \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) & 0\times \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \end{array}\right)=\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right).
$$
Once you have all of this, you simply multiply it out:
$$
\langle 01|X_1X_2|01\rangle=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \end{array}\right)\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)\left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right)=0.
$$
Shorter Method
(With experience, this method lets you perform this calculation just by looking at it! Of course, when giving an answer, I don't recommend that; you should always justify your answer.)
Remember to think of each term $|01\rangle$ as $|0\rangle\otimes|1\rangle$. So, when you also write $X_1X_2=\sigma_1\otimes\sigma_1$, you see that
$$
(\sigma_1\otimes\sigma_1)|0\rangle\otimes|1\rangle=(\sigma_1|0\rangle)\otimes(\sigma_1|1\rangle)
$$
i.e. when everything is just tensor products, individual terms match up. Now, hopefully you know the action of $\sigma_1$ and $\sigma_3$ on the basis states:
$$
\sigma_1|0\rangle=|1\rangle\qquad \sigma_1|1\rangle=|0\rangle \qquad \sigma_3|0\rangle=|0\rangle \qquad \sigma_3|1\rangle=-|1\rangle
$$
Thus,
$$
(\sigma_1|0\rangle)\otimes(\sigma_1|1\rangle)=|1\rangle\otimes |0\rangle=|10\rangle
$$
One can then easily observe that a state such as $(|01\rangle+|10\rangle)/\sqrt{2}$ is acted on by $X_1X_2$ to give
$$
X_1X_2(|01\rangle+|10\rangle)/\sqrt{2}=(|01\rangle+|10\rangle)/\sqrt{2},\tag{1}
$$
the same state. So it is clear that the inner product of the state with itself is 1:
$$
(\langle 01|+\langle 10|)X_1X_2(|01\rangle+|10\rangle)/2=1.
$$
On the other hand, had the outcome in Eq. (1) been a different one of the four Bell states, because we know the Bell states form an orthonormal basis, the expectation value would be 0.
