I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant theory regarding the blow up of a compact symplectic manifold along SEVERAL compact symplectic submanifolds.
In general, I am seeking to know more about how one $\textit{sequentially}$ blows up a compact symplectic manifold, obtaining a sequence of maps $$\tilde{M}_{n}\rightarrow \tilde{M}_{n-1}\rightarrow...\rightarrow \tilde{M}_{1}\rightarrow \tilde{M}\rightarrow M$$ How does one, explicitly do this? If I recall correctly, away from exceptional divisors (more generally, the preimage of the center of the blow up map), one has symplectomorphism, so does this mean that it is preferable to always take the next center of blow up to be symplectomorphic to a submanifold in $M$ that is mutually disjoint from the first center? That is, if $\tilde{X}_{k}$ is the submanifold of $\tilde{M}_{k}$ along which blow up takes place and $X$ is the submanifold along which the first blow up in $M$ takes place, then one should always choose $\tilde{E}_{k}$ to be symplectomorphic to a compact symplectic submanifold of $M$ that is mutually disjoint from $E$? Does it matter?
If so, then what important things are there to know about blowing up $M$ along, say, 3 symplectic submanifolds? What happens if they are mutually disjoint in $M$? What happens if they intersect? If the intersections are transverse, what effect does that have? In the task of locating a given number of mutually disjoint symplectic submanifolds, are there some standard problems to confront, i.e., are the submanifolds size-restricted? What other restrictions are there? How does my search for such submanifolds change when I apply the constraint that I am seeking compact symplectic submanifolds that have the Hard Lefschetz Property (that their symplectic form $\omega$ makes an isomorphism $L_{[\omega]}:H^{n-k}_{dR}(X)\rightarrow H^{n+k}_{dR}(X)$ for every $0\leq k\leq n$).
I have a few resources for study at my disposal (algebraic geometry texts by Griffiths&Harris and Fulton, as well as standard symplectic topology research papers by Cavalcanti and McDuff) but either my understanding is currently too primitive, or I am correct in noting that the answers to my questions are not all to be found together in a comprehensive source. Perhaps some of them haven't even been yet established in the literature.
Can any of the algebraic geometers and symplectic topologists here help to either point me towards additional resources that are detailed and not too obfuscatory and that don't depend on a couple of years of algebraic geometry knowledge (I am more on the symplectic side myself and am only acquainted with some beginning graduate level algebraic geometry), or generously offer some of their factual knowledge and helpful intuition on my questions?