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I would like to get some references where I can find the theory of the real representations of $\mathbf{SO}(n)$ and $\mathbf{U}(n)$.

In particular, I would like to know for which dimensions there exist irreducible representations of these groups and for which dimension, how many non-equivalent representations are?.

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Take a look at John Frank Adams' Lectures on Lie groups. W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+182 pp.. chapter 6.

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Read about Frobenius-Schur anywhere. In a nutshell a complex irreducible $V$ of complex dimension $n$ can give 1 or 2 real irreducibles, whose real dimensions are $n$ or $2n$. This can be easily determined by computing the FS-indicator or dimensions of invariants in both $S^2V$ and $\Lambda^2 V$. The latter can be done in Lie for each particular representation.

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    $\begingroup$ Minor correction: $n$ or $2n$, not $n$ or $n/2$. $\endgroup$
    – David E Speyer
    Commented Apr 2, 2020 at 14:55
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The book by Broecker and tom Dieck (Representations of Compact groups, Springer Graduate Texts in Math) has a very useful section about real, complex and quaternionic representations and how to pass between them.

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