I am making a batch of about 500-1000 PCB's, which option is cheaper,
You can find this out "easily" by getting quotes from your vendor both ways, and also pricing out the components from whichever vendors you'll purchase them from.
Typically when ordering turnkey, you will be charged a percentage markup from the cost of the parts to pay for the work of ordering the parts, sorting them out, etc. Some vendors will be more and some less transparent about what they are paying for the parts and where they are buying them.
which option is more reliable,
Kitted assembly reduces the risk your manufacturing partner will buy counterfeit parts or just order incorrect parts due to a misunderstanding.
and how do I let the company know which part to put where?
You will probably need to supply your vendor with an assembly drawing showing the part locations visually and an "X-Y" or "pick and place" file giving the location and part number for each part.