Is there a right way and a wrong way to setup 301 redirects? Am concerned because after 10 days I am not seeing my new URL's achieve anything close to the SERP my old URL's achieved.
Server side 301 redirects without any jumps are the appropriate method. You can check response cods from servers by using live http headers or another such utility. Also, 10 days is possibly not enough time for Google to digest everything. If you want me to take a closer look then pm me the old URL, the new URL, and a phrase the OLD URL formerly appeared in the SERPs.