What is an example function of a transitive yet non-reflexive and non-symmetric relation?

How about $f(n)=17$ for all $n$?

Suppose $f$ makes $R_a$ transitive. Let $y,z\in \mathbb{N}$ such that $$f(x)=y, f(y)=z$$ for some $x \in \mathbb{N}.$ Transitivity now implies $f(x)=z.$ So $y=z.$ So $f(y)=y$ for every $y \in f[\mathbb{N}].$

Of course, constants are an example. But they are not the only example. The above argument suggests that you only need to find a function which fixes the elements in the range. For instance

$$f(n)=\begin{cases}1, \text{if }n \text{ is odd}\\2, \text{ if }n \text{ is even}\end{cases}.$$

$f$ is not reflexive because $f(3)\neq3$ and not symmetric because $f(3)=1$ but $f(1)\neq 3.$

Obviously you could have chosen any odd and even number in place of $1$ and $2$ respectively.

You can come up with plenty of other examples in a similar way.