For background, reference these questions:
How can I increase the density of residential zones?
How do you provide more workers to your city?
Ok, now that the basics are covered, let's talk about purpose of R, C, and I. in a town with a manufacturing specialization.
R is the source of your residential agents. If you are focusing on a specialization, your main interest in R is that it brings in workers. Place parks to set the land value of an area and force a given wealth level to settle there. All R wealth levels start at low density and expand as happiness accrues. You will need areas that are not near parks for low wealth to settle into and expand into LW HD R buildings. You want to keep your jobs full, so add (or density up) more R until that happens. Workers of a given wealth level will only work at jobs with a matching wealth level (most buildings have jobs for each wealth level).
You will want at least the LW and MW residential buildings, as they supply the most workers per building. If you are having tax trouble and have space to work with, then add HW residentials.
C is primarily a source of R happiness, which will cause R to increase in density. Usually you'll want to satisfy at least half of your shoppers. Shoppers of a given wealth level will only shop at C buildings of the matching wealth level. C also provides jobs and taxes. You may want to keep C at medium density for tax and traffic purposes. C mainly increases in density by selling its goods. If you overbuild C, you'll have extra jobs pulling workers away from your specialization and extra unsold goods which will cause C to abandon.
If you need to increase R happiness and really can't spare the workers for C, use parks (with expansions). The "Medium Field with Parking" is ideal low wealth R in size, capacity and cost. It does not increase land value, so you can place it near your LW R without worry.
I is primarily a source of jobs without an increase in goods. Since you have a specialization, it's entirely possible you don't have spare workers for I to soak up and therefore do not need any I buildings.