The topic police might close this, but this is actually a very reasonable question.
How do I find manufacturers for the different components (PCB, getting
it soldered together, case, etc.)? Or maybe it's better put to say
that I'm trying to find out how to shop manufacturers.
These are Contract Electronics Manufacturers (CEMs)
Without knowing where you are, I cannot help with specific companies, but I would expect there are some reasonably close. These companies usually have their own fabrication (PCB) vendors and supply chain kitting agents that they use and that will normally get you the best price.
Should I split manufacturing to try to prevent IP theft?
It will cost you more (typically) if you do. It is not uncommon for an NDA to be put in place. It is in the CEMs best interest to keep your design confidential (word gets around if they do not and that means they lose business).
Does the shop that assembles the boards also program them?
If you tell them how to and provide the programming files. You will be charged for the service as part of a larger quote.
Depends on where you intend to ship the units to. Europe has CE, USA has the equivalent (different versions of the same tests), as does Japan, Australia and New Zealand. You should look up the required certifications for your region.
The rest is determined by what you have and where you are.
Based on the discussion in meta, here are the various steps that have to be taken to get a product actually to market, each a valid question in its own right.
PCB manufacture. Most CEMs use their own sources as I noted, but you should ask for Design for Manufacturability comments. The PCB fabricators know what they can reasonably produce and can give helpful feedback to make the design better for them (and usually less expensive for you). This alone is a huge subject.
PCB Assembly: This is where the assembler can give likewise helpful comments (device spacing and host of other things). Once more a large question.
Design for test. This is also a question for the assembler; you would usually want them to test the device and the assembler knows what testing methods they prefer (there are many).
Programming; It may be that you have a suitable programming solution, or perhaps you can get the devices pre-programmed by the IC distributor (they will for a price provided the volumes are high enough) or you can ask the CEM to do the test. This is usually a quite long discussion.
Housing manufacture and product assembly. Sometimes the CEM will offer this service, and they can again answer questions about the most efficient method to achieve this.
Harnesses; there will be connectors and often cable harnesses which will need to be made; you could investigate your own sources or ask the CEM for suggestions.
Regulatory issues; I have mentioned some standards, but the entire area of compliance is complex and has many gotchas where experts really need to be asked their opinion. What are the regulations in my area and what does it take to pass them might be a good starting question here.
For the USA, you need to be wary of the fact that all 50 states (and many localities within them) have their own rules that may or may not impact your product. Once more, a discussion in its own right.
Obsolescence: A nasty word but one that is all too real in electronics - how would that be managed? (there are numerous methods including the CEMs who have subscriptions to parts intelligence databases).
Design for Bill of Materials. You may be buying a few bits from Digikey and co, but the CEM will want to buy from primary distribution and they will want to use reels, not a cut tape (for small devices) and tubes or trays for larger devices as their lines are designed for those things. You do not want the shock of finding that a diode you would like to use (or works in your prototype) has a minimum order quantity (MOQ) in distribution of 3000 parts on a reel. Another discussion in its own right.
There could be more but hopefully you can ask all the necessary questions; if nothing else it would be educational for both yourself and others.
Last but not least: the documentation pack. When I send a design out, the pack contains literally everything necessary to produce the design. This includes a master assembly drawing, a master PCB fabrication drawing (which contains information not in the gerber / ODB++ electronic data), drill drawings, test requirements, test methods with pass fail criteria, schematics (if you want the CEM to troubleshoot failed items), full production BoM (with all non populated parts removed).
Clearly, gerber / ODB++ data is sent as well (all the stuff necessary for the fabrication to be done). ODB++ is the preferred method for the better assemblers as of 2018.
My master assembly print also lists the identifiers of not fitted parts as they are usually still in the pick and place file (there are ways of ensuring they are not depending on the CAD tool used); if they are still in there, it is difficult for the CEM, so you need to help them out.
The documentation I provide meets IPC-D-325 incidentally.
I am not going to even get into export regulations.
Here is a (slightly redacted) fabrication master front page (page 1 of 3):