I got my PE in '95 and started moonlighting a couple years later. Three years ago my employer decided this was not allowed. At that time I was getting 1/3 to 1/2 of my income from moonlight work, so I quit them and went solo full-time. I'm now earning triple what I made then, and still working the same 60 hour week.
I highly recommend the moonlight strategy if you can get away with it. Starting up is hard enough from a sales and service delivery standpoint. Throwing in all the tax, insurance, and other business issues reduces your odds of success. It's best to tackle those AFTER you have a satisfied customer base in place. And there's no safer way to build that base than under the security of someone else's payroll.
Start with cold calls to a few local architects and HVAC contractors who do small design/build projects. Spend way too much time on your first few projects and get them perfect. Then charge slightly less than market rates. The trick is get your systems and CAD standards in place so that projects later on go seemlessly and fast. That's how you build a reputation and make money.
Be very certain that you don't short-change your current employer by moonlighting. Maintain definite boundaries between the two worlds, and err on the side of over-serving the folks paying your salary. I actully found that my day-job performance went up as a result of moonlighting. I learned to be more self-reliant and time-conscious. It turns out that while I was learning to how to make more money for myself by being more focused and effective, I was also making more money for them. They really should have made me a partner instead of a competitor. Oh well, it's their loss.
Regarding competitor vs. subcontractor, do both. Most folks aren't as concerned about the details of the business relationship between two organizations as they are with getting the work done better, faster, and cheaper; and these are the only features that matter. Every pitch you make to a client will boil down to at least one of them.
Specific service needs that seem to have arisen recently include:
Design/Build Documents - used to solicit bids from D/B contractors for Mech/Elec/Plumb (MEP) projects. Can also be used for permitting. Architects use this strategy to keep their fees low in highly competitive situations by not doing detailed designs for these systems.
Permit Documents - contractors usually don't have PEs on staff and need a stamped dwg to get through plan review. They want fast and cheap, but you can leave much of the non-critical details up to them.
Full Plan & Spec preparation for large MEP projects will likely be a few years away. This is the preferred arena of the large firms you will ultimately compete against due to the large fees involved. It is also the highest risk work you can do, so don't go there without insurance and impeccable work-product. Architects will be reluctant to team-up with you on these until you have a reputation established.
One final note on marketing. It's sort of silly, but after I was in business for a year, I sent a holiday card with my business card attached to every architect and contractor in my area. It cost me about $1 each, less than $500 total, and I got over a dozen new clients from it. Cheapest advertising I've ever seen. Good Luck.