The link in Mario Gianota's comment is not the official JLCPCB  website please don't put any sensitive information into that website. https://jlcpcb.com/ is a reputable place to go and has a good community behind it like the educational youtuber that put me on to them "GreatScott!" (Who is a spectacular resource when learning about electronics and if you're in this Stack I assume his content will be right up your ally). I wish you the best of luck in your projects!

I get my boards made by JLCPCB.com and I assemble them myself. I wouldn't pay money on top of board production to get someone to assemble the board unless I was doing a large volume production.

I use Schotry International in Beijing. Their quality is good, turn time about 45 days for new orders, and cost is usually slightly above the BOM cost. But they do charge a $300 NRE and don't like to do small (~10 orders) so this is not a good option for 5s and 10s. But great for 100 and up.

I am a recent convert to JCLPCB and have tried assembly service with great success. I had them load about 80 SMT components from their own stock on 5 boards and it cost about AU$40 including the components they used. The process added a couple of days to PCB fab time, and the component cost was much lower than Digikey etc. Design files are generic Gerbers and drill file along with a modified pick and place file (also from pretty much all PCB design packages). All steps of the way are supported by the company in terms of pictorial quality control checks as well as good tutorials. They even fixed some of my component orientation mistakes without charge. Most impressed with the whole process flow. It's worth going through the process with a test project just to see how it all works and in the end you don't even need to place the order.

Wow, that's super cheap!  Assembly prices I usually see are at least 10x that for such low quantities.  I use JLCPCB or OSHPark depending on what kinds of boards I want, and I order from LCSC too.  I had no idea they did assembly too.

I really want to try out the Aisler assembly service one of these days.
It's a fairly expensive service compared to other options, but it uses EU and American-based manufacturers instead of chinese, which I feel a bit more comfortable with.

I never used the full assembly but their PCBs have always been of top quality. 
Assembly specs are "single sided only", which is a bummer but can be dealt with. My "LZRTag" PCBs, which have about 25€ worth of components on them and a few more € of PCB, cost about 50€ per piece for ten pieces of assembly, with lower prices as you scale up. Again, a bit pricey, but fully EU based, with the full part catalog of DigiKey and co.

Minimize the PCB size by using BGA and 0402 components or similar, reducing the number of individual components, and I can imagine it to be a good option :>

OK, I misunderstood your question.  I also get PCBWAY to do my assembly once I move past prototypes.   It takes about 3-4 weeks if you have them do the part ordering and everything.  Naturally it gets cheaper with higher volume.  Be careful about the final cost thought.  The US currently has a 25% import duty on China goods over $800.  I discovered the hard way that this includes PCB assemblies.  Great for my customers :(

Thank you very much for your info, it is appreciated :)

For my own SMD prototypes it's actually pretty easy.  Get a tube/pot of solder paste, and use the stencil to spread paste on the pads (just needs a flat surface and a credit card to do the screening).  Then use some fine tweezers to place the chips.  I'm 62 with bad eyesight, and I can do down to 805 components. Then put the PCB in a toaster oven (or even a conventional oven) set to 400-450 and wait about 3-4 minutes.  Do a final inspection and fix any imperfect parts.  I recommend getting a ChipQuik rework kit for tools and materials.  Here is an example... the green PCB  https://hackaday.io/project/170932-hoverboards-for-assistive-devices/log/182221-log-12-the-blujoy-bluetooth-joystick-interface

I use Eagle cad to design PCB's and then get them made by PCBWAY.com.  For small prototype runs it's super cheap ($5 for 10) and shipping is reasonable considering you get them a week later from china.  I never make my own soldered proto-boards any more.  I go from plug-in proto board to PCBWAY PCB board, with at least a couple of revisions.  Plus designing a PCB is a valuable skill to learn.  For a few extra dollars they will include a stencil, so surface mount is definitely doable as well.

What about PCB assembly?  Placing the components on the board and reflowing them?