Hi guys! This is a really dumb question I'm sure, but I am just beginning to get into this miles stuff.

I am signed up with Delta Sky Miles and currently have about 44,000 miles from using my Amex card. I'm ready to book my first flight to earn qualifying miles, but I am wondering how this works with partner airlines. Sometimes Delta is really expensive and I'd rather fly a partner airline at a cheaper rate.

How does this all work? Could I book by finding a flight on a website like matrix.itasoftware.com? Or when they say "partner" flights do they mean only the flights that show up on Delta.com but are listed as being operated by another carrier?

This is all so confusing!

EDIT: Sorry! I misspoke. I have the Amex card--not Chase. When I wrote this I just got finished reading about Chase in another window. Yikes!