If I consider $a|00\rangle + b|01\rangle +c|10\rangle +d|11\rangle$ as a valid two-qubit system, what is the probability of measuring the first qubit in state $\frac{1}{\sqrt 2}(|0⟩+|1⟩)$?
I know that the probability of measuring the first qubit in state $|0\rangle$ is $|a|^2 + |b|^2$, but I couldn't figure out how it works here. Thanks in advance.
Given an arbitrary state $|\psi\rangle$, if it is expressed in the computational basis as $|\psi\rangle=\sum_k c_k |k\rangle$, then it will give the $k$-th result (when measuring in the computational basis) with probability $|c_k|^2$.
Note that here by "computational basis" I simply mean the measurement basis under consideration.
If you consider another type of measurement, corresponding to a different basis, say $\{|u_k\rangle\}_k$, then to figure out the outcome probabilities in this new basis you need to express $|\psi\rangle$ in terms of the $|u_k\rangle$. Say that doing this you get something of the form $$|\psi\rangle = \sum_k d_k |u_k\rangle.$$ That means that the outcome corresponding to $|u_k\rangle$ is obtained with probability $|d_k|^2$.
If you start with a description of the state in the $|k\rangle$ basis, and want to switch to a description in terms of the $|u_k\rangle$ one, you simply need to compute the inner products $\langle u_k|\psi\rangle=d_k$.
More generally, given a state $|\psi\rangle$, the probability of finding it in a state $|\phi\rangle$ is given by $|\langle\phi|\psi\rangle|^2$.
There is, however, a slightly more general way to do measurements, that doesn't (necessarily) involve a full collapse of the state. You can ask questions of the form "is the state in a given subspace?". For example, you can ask whether a state is in one of first $3$ (or any other subset of) computational basis states. You model this situation by using a projector $P$ that projects onto the required basis, and the associated measurement probability is then given by $\|P|\psi\rangle\|^2$.
Taking as an example your specific case, you have a bipartite state expressed in terms of the computational basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ with coefficients $a,b,c,d$. Let's call this state $|\psi\rangle$. You are asking about the probability of the state being found in the subspace $\{|+,0\rangle, |+,1\rangle\}$. To compute this probability you therefore write down the projector $P$ defined as $$P = |+,0\rangle\!\langle +,0| + |+,1\rangle\!\langle +,1| \equiv |+\rangle\!\langle +| \otimes I_2,$$ and you compute $\|P|\psi\rangle\|^2$. Equivalently, you just rewrite $|\psi\rangle$ expressing the first qubit in the $|\pm\rangle$ basis, and then sum the probabilities of finding it in either $|+,u\rangle$ or $|+,v\rangle$, with $\{|u\rangle,|v\rangle\}$ the basis chosen for the second qubit (it doesn't matter which one is used).
