In the following every surface is assumed to be connected.
I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative monoid of homeomorphism classes of compact surfaces (with boundary)? Does it also have a nice presentation? I think it is generated by the $P[k]$ and the $T[k]$, where $[k]$ means that $k$ holes have been inserted, and $k$ runs through the natural numbers. What relations do we need? And how do you prove that no others are needed? 


Another question, which is rather informal: Do you think that it's worth to read the proofs of these classical classifications? I know the importance of the results, but I suspect that the proofs are just technical.



Presentation of the monoid of surfaces
I assume you want your surfaces-with-boundary to be compact? Anyway, this cannot be generated by the $P[k]$ and $T[k]$, since you are leaving out the genus 0 surfaces (spheres with holes). Since connect-sum-with-a-disk is the same as removing an open disk, I would work instead with the generators $P$, $T$, and the disk $D$; the $D$ won't interact with the other generators. Since every surface-with-boundary is a surface-minus-some-disks, it seems a presentation is $\langle P,T,D\vert P^3=PT\rangle$.


In answer to your second question: it can be very worthwhile to know the proofs of these classical facts, but that doesn't mean you need to learn the classical proofs. For the classification of surfaces, I have seen Benson Farb give a very nice proof (fitting with the "modern" perspective on mapping class groups etc.) hinging upon the fact that the sphere has the maximal Euler characteristic among surfaces. If I can find any notes of that lecture or a written version, I'll update with a link.


