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 without proper padding scheme. For example, if you use $e=3$ without a proper padding scheme than you will be vulnerable to cube-root attack.
- For encryption, you can use PKCS#1 v1.5 padding or Optimal Asymmetric Encryption Padding (OAEP), Prefer OAEP, PKCS#1 v1.5 has many attacks and hard to implement correctly.
- For signatures, you can use Probabilistic Signature Scheme (PSS).
And note that RSA Signing is Not RSA Decryption!
