You should use multithreading as well as pipelining requests. For example search->details->save
The number of threads you can use doesn't depend on your equipment only. How many requests the service can serve? How many concurrent requests does it allow to run? Even your bandwidth can be a bottleneck.
If you're talking about a kind of scraping - the service could block you after certain limit of requests, so you need to use proxies or multiple IP bindings.
As for me, in the most cases, I can run 50-300 concurrent requests on my laptop from python scripts.