I have built a company for designing proteins for use in therapeutic synthetic biology (CRISPR/CAR/SynNotch/etc. systems). My undergrad background was more chemistry/physics/philosophy while my graduate school was in biochemistry/cell-bio, and I think that accidentally set me up well to work in synthetic biology. The books have yet to be written :-)

The rules in synthetic biology are sloppy versions of the rules found in chemistry and physics. The logical/statistical/mechanistic tools of the the more mathematical sciences are extraordinarily useful to have in mind, however you have to get comfortable with a lot fewer 'correct' answers and a lot more unknown or even unknowable variables. Some tools like population level statistics are really useful tool in biology, while complex logically sequential steps used in programming really don't work so well in the wet world of biology. Synthetic biology is very much parallel programming of thousands of 10-line programs, rather than 10, thousand-line programs. To me it seems the hardest part for people who haven't done biological lab work to understand seems to be the intuition for how biological proteins interact with each other, and the scale of their interactions in time and space, both upwards and downwards - relative to the cell itself, as well as the chemistry involved in the cell. That intuition at the 'meso-scale' is uncommon, and building it without actually running experiments in lab is tricky.

It's still a new field, so it's not particularly well-articulated in terms of sub-fields right now. But there are huge differences in terms of whether one is studying prokaryotic systems, single-cell systems, or mammalian systems. Or whether one is manipulating proteins, DNA itself, or biologically compatible materials. Or whether you're building academic sensors, commercial systems, or therapeutics. Though they all involve overlapping ideas, the edges to each system are really pretty different to my mind. Another tricky thing for those without the biological background is to be able to judge whether a synthetic-biology tool is useful in the context of biology (and not just a toy).
What is a cool and useful trick for a digital computer or an physical robot can often be either trivial or impossible in a cell, while engineering a cell to biologically integrate a novel sensor can be an amazing breakthrough. Nano-scale metal gears, robots and antennas with massive energy reserves just don't make sense in the warm, wet, energy-efficient, Brownian world of biology.

iGEM is a great place to start - it's an academic competition for undergraduates - a lot of exploration going on there that is relatively accessible, if a bit unrefined. [1]

Our company, Serotiny, is trying to bring a plain-language understanding to the synthetic design of novel proteins, which are often the payloads and tools of synthetic systems like Cas9 in the article. [2]

Addgene is a non-profit physical repository of many of the genetic 'tools' used, and they have some nice blog-posts and tutorials for beginning scientists. [3]

SynBioBeta is the only real industry group around right now for the field, and they keep pretty well up-to-date with interesting industry news - new companies, new products, new events related to synthetic biology. [4]

[1] http://igem.org

[2] https://serotiny.bio/notes/proteins

[3] http://blog.addgene.org/

[4] https://synbiobeta.com/