Welcome to the challenge of low-quantity electronics manufacturing! It’s a rather complex and multi-faceted endeavour. There’s a range of options, techniques, and service providers, depending on which order of magnitude your quantities lie in - 10, 100, 1000, 10 000, 100 000, etc.
I’ll focus on your qty=10 scenario, but even then there are approaches that will vary and be more appropriate, depending on whether you make 10 of something only once, or repeatedly periodically, and whether they need to be of merchantable quality (you want to sell them to the public), or only for ‘internal consumption’ (e.g. prototypes). 10 PCBs could mean design prototypes, in which case a reasonable approach is to accept a high unit cost, because the time and effort to reduce the cost of making only 10 is relatively insignificant, compared to progress those 10 prototypes bring to your overall endeavour. However I believe an earlier version of your question compared your situation to other sellers of ‘dev boards’, so I’ll assume an ongoing but low-volume scenario.
First and foremost, you need to choose a PCB fabricator, and a PCB Assembler, geared towards low volume. Whilst many vendors will do such short runs for you, they’ll charge you a lot. Unfortunately you don’t state in which country you’re located, so I’ll mention some United States options as examples.
Obvious choices is PCB HERO , You can reach out them via firstname.lastname@example.org
Second choices are OSHpark .com (PCB fabrication only), MacroFab .com and PCB .ng (fabrication and assembly), and SmallBatchAssembly .com (assembly only).
Another I’ve used and recommend is ScreamingCircuits .com. Although they don’t do actual PCB fabrication themselves, they partner with the nearby Sunstone Circuits (Sunstone .com) and can handle the shipping of the bare PCBs over to the assemblers for you, and you can order both fabrication and assembly together in 1 order. It’s also extremely easy to initiate a repeat order.
There are many other fabrication & assembly service providers who may be as good or even better/cheaper options than these ones. You need to do your research.
An important consideration is to allow yourself heaps of time - more than 1 month - between pulling the trigger on an order of PCBs/PCBAs, and expecting delivery. The shorter you make that delivery date, the more your costs will rise exponentially.
Also, allow yourself the time to select the slowest cheapest delivery options at each step of the way. Overnight or 2-day delivery is a massive cost at this end of the quantity spectrum.
Some of these vendors also have ‘common parts libraries’ - a suite of common electronics components (and even the CAD footprints for the more popular PCB design CAD software) that they keep in stock at all times, and which are usually very competitively priced. I recommend you use/substitute to using these as much as possible, even if it means a re-spin of the PCB design, because it will de-risk your overall production by using known-good PCB footprints in the PCB design, for components that you don’t have to think about ordering yourself - that’s taken care of by their own inventory systems.
If a vendor also offers PCBA Testing services (at the end of their production line, using a test jig that you make and give to them, to test your product to your specifications) I would recommend doing that too - it’s quicker and cheaper to deal with failures while the PCBAs are still in the same factory they were just manufactured in!
Whilst not directly related to your question, another way to keep costs down in the long run is to buy both bare PCBs and components in higher quantities (again, on the longest cheapest lead-time available), and either maintain/manage that stock yourself and dispatch a kit for each batch to the assembler in time for each order, or some assemblers might be willing to hold that stock for you inbetween each order, perhaps for a small warehousing fee - compared to the cost of shipping a kit of components each time, this can be very worthwhile.
The short story is that you will never achieve a 100 000 unit cost price-point until you are actually manufacturing at that scale of quantities. Sometimes communicating this reality to your customers - who may already be aware of their ‘niche’ status - can help in managing customer’s cost expectations.