It would help if you would add more photos from different angles.

You aren't going to get any useful advice on-line. Call a pro or two and talk it over and get quotes on the options. It's pretty clear to me it's not a DIY project.

Agree with Sigrid. Usually this requires some type of terracing— broader areas if more walkable space is desired. I’m no pro so I have no specific ideas- but just to expand on your - there would be all sorts of issues about wall support, fill, drainage, access, and if there are different options, would have different costs and other pro/ cons and different results as to what is where.
You could Google or use Pinterest to see some before/ after solutions for sloped yards, not for take- it- to- the- bank, but to see if anything looks like what you might have in mind, though not to say, “I want just like this”, because you want input, from landscape architect, same as with a house. Also same as with house- building, you might noodle on a list of desired uses for any “ more lawn” and what is prompting change, like, is patio too small for dining, is slope too steep for kids to enjoy, do you prefer gazing at level space, is lawn slope hard to mow, etc.

You have to meet the grade or put up a retaining wall. There's no magic trick. Retaining walls need to be properly engineered with lots of prep so they have the right base and don't shift. Equipment will need access to dig for that.