> But what is the most appropriate choice for it?

For public exponent $e$, small values are preferred like $\{3, 5, 17, 257, \text{ or } 65537\}$. With this, we can guarantee that the number of operations is low. We can control this with our choice. Of course, for the choice of $e$, we must have $\gcd(e,p)=1$ for any prime $p$ divides the modulus $n$. This guarantees that we have the inverse of $e$ such that $e\cdot d = 1 \bmod \phi(n)$, and $\gcd(e',n) = \gcd(e,n)$
 
> Should it be small compared to $\phi(n)$ or approach it?

You can choose a public exponent $e'$ bigger than $\phi(n)$, however due to the congruence, we can always find an $e$ such that $ e' \equiv e \bmod \phi(n)$ with $e < \phi(n)$.

Of course, RSA should never be used with **proper padding scheme**. For example, if you use $e=3$ without a proper padding scheme than you will be vulnerable to [cube-root attack][1].

 - For encryption, you can use [PKCS#1 v1.5][2] padding or [Optimal Asymmetric Encryption Padding][3] (OAEP), Prefer OAEP, PKCS#1 v1.5 has many attacks and hard to implement correctly.
 - For signatures, you can use [Probabilistic Signature Scheme][4] (PSS).

And note that **[RSA Signing is Not RSA Decryption][5]!**


  [1]: https://en.wikipedia.org/wiki/Cube_attack
  [2]: https://en.wikipedia.org/wiki/PKCS_1
  [3]: https://en.wikipedia.org/wiki/Optimal_asymmetric_encryption_padding
  [4]: https://en.wikipedia.org/wiki/Probabilistic_signature_scheme
  [5]: https://www.cs.cornell.edu/courses/cs5430/2015sp/notes/rsa_sign_vs_dec.php