You prob know this, but just in case: Google Flights (https://www.google.com/flights/) is a terrific resource for finding flights at the best price, esp. if you’re able to flex your departure and return dates somewhat.
We fly Delta (and its international partners) almost exclusively, but I prefer Google Flights to the Delta website for exploring itineraries. You can narrow the search to specific airlines/partnerships, # stops, flight duration, etc. I then book the flights through the airline website.
As for lodging, we generally prefer to establish a base city when we travel internationally and much prefer an apartment to a hotel. In addition to providing a “live like a local” experience, it saves lots of money–not only on the lodging but also on meals. We much prefer a leisurely breakfast in an apartment to heading out to a hotel lobby or restaurant.
And we use public transit almost exclusively–not only a lot cheaper but also often faster.