Appropriate public exponent choice for RSA encryption From what I understand, we can arbitrarily choose an exponent e as long as 
 $\gcd(e,\phi(n)) =1$.


But what is the most appropriate choice for it?
Should it be small compared to $\phi(n)$ or approach it?
we can arbitrarily choose an exponent $e$ as long as $\gcd(e,\phi(n))=1$.

No. If we want any security, we further :

Must NOT choose $e=1$, because that makes $x\mapsto x^e\bmod n$ the identity function over $[0,n)$.
Must NOT choose $e$ in a way revealing information about of $\phi(n)$ or the factors of $n$ beyond that $\gcd(e,\phi(n))=1$. For example, we can not choose  $e$ as $e=p$, nor $e=\phi(n)-3$, nor as the first integer larger than $\phi(n)/42$ with $\gcd(e,\phi(n))=1$, nor in such a way that $d$ is small.
Should choose $e$ not too small if there is a regulatory minimum or we have little assurance about how the key will be used. This mitigates to some degree some poor message padding and some poor decryption implementations. Using $e\ge2\log_2(n)$ should be technically OK for all except the worst (or absent) paddings.


What is the most appropriate choice for $e$?

The simple, safe, standard option is: choose $e=F_4=2^{(2^4)}+1=65537$, then choose prime factors $p$ of $n$ with $p\bmod e\ne1$. This test (and factors being distinct of $e$, which holds for suitably large factors) is enough to ensure $\gcd(e,\phi(n))=1$, since we picked $e$ prime. And picking factors as a function of $e$, rather than the other way around, ensures minimal information is revealed about $\phi(n)$ and factors of $n$. $65537$ is large enough that $e\ge2\log_2(n)$ is met for practical sizes of $n$, and conforms to recommendations by all major security authorities.
If for some reason we must choose $e$ after factors of $n$, a common practice is to choose the lowest $e$ above some minimum with $\gcd(e,\phi(n))=1$. A very slightly safer practice would be to choose $e$ randomly in some interval $[m,2m)$ until $\gcd(e,\phi(n))=1$, but that's overkill (and there's seldom a good reason to choose $e$ after the factors of $n$ anyway).

Should $e$ be small compared to $\phi(n)$ or approach it?

The former: $e$ should be small, and $\phi(n)$ large. There is often a conventional upper limit to $e$, like $e&lt;2^{256}$ in FIPS 186-4, or $e&lt;2^{32}$ in some Windows APIs, when $\phi(n)$ must be in the thousands of bits. And for reasons already spelled, $e$ must not be chosen close to $\phi(n)$. Also, it is best to keep $e$ not too large for performance reason: public-key use requires time roughly proportional to the bit size of $e$. The standard $e=F_4=2^{(2^4)}+1$ is attractive because encryption with it is faster than with larger values of $e$ (and as well as most lower values). Still, that $e$ makes public-key use about 8 times slower than with $e=3$; the latter can be a good choice when the performance of public-key use is paramount, appropriate padding and implementations are used, and regulatory requirements allow.