I would run long sprinkler hoses (those flat perforated ones) for the narrow 10' swath on the top side-yard of the diagram. It will run from the patio to the front-end of the house.
On the 38x30 front section on the right and on the 36 x 50 back section on the Left, I would run whatever sprinkler you would like according to your budget.
I would opt for two oscillating sprinklers myself, with the adjustable width and how far they oscillate back and forth, so you don't water what you don't need to.
I doubt you want to run a hose from the driveway bib to the lawn so I would get a 2-nozzle automatic timer that will run the left/backyard side and the side-lawn sprinkler hose separately at automatic intervals. Then a single nozzle automatic timer for the right/front yard side programmed to run at a different time than the others.
You can get all sorts of automatic timers, smart wi-fi capable, Bluetooth, or digital clock timers. I don't have a recommendation but will state that I use the "Aqua Joe AJ-ET2Z Easy 2-Zone Electronic Timer", and have had a few good trouble free seasons with it. I also like an old-fashioned "Orbit 62041 Metal Mechanical Watering Timer" as well. You could get a few of those and a y-splitter for your bibs to handle the different areas. I only water once a week so manually timing them isn't a big issue.
I recommend hose reels to help pull them up when time to mow.
Those perforated sprinkler hoses are a little bit of a PITA to take up and lay back flat to where they have good coverage so someone may have a better recommendation for the long narrow side yard. I'd guess the newer mini adjustable oscillating sprinklers that can be ran in sequence could be a good fit for the side-yard.

 

2x 100ft flat sprinkler hoses down the 10ft x 74ft+30ft section.
1x eden or melnor oscillating mini turbo spike sprinkler on the 38ft (-10ft covered by the soaker hose) x30ft section.
1x eden or melnor oscillating mini turbo spike sprinkler on the 21ft x15ft section.
2x eden or melnor oscillating mini turbo spike sprinkler on the 36ft x50ft section (each on a separate hose and timer). Maybe add a 50ft run of flat sprinkler soaker hose if you got low flow rate and can't cover the area fully with the eden/melnors.
3/4" Gilmour Pro Contractor hoses (full port connectors on the Gilmours) end to end. On the 38FTx30FT area, you might be able to get away with some cheaper Worth hoses (Worth hoses are not full port connectors).
For splitters, you'll likely have to pay attention to the port size, and search for the ones with the largest ports (full flow) to get the largest coverage area per sprinkler possible. The ugly silver/gray plastic ones from STYDDI are barely ok, but not quite full port. If they don't work, other option is to DIY some with 3/4" PVC (use brass FHT to FPT for your female connector inlet, the plastic ones break).
Run one sprinkler or 100ft length of sprinkler hose at a time for max coverage.

Overall, thats an easy area to cover. Solenoid valve timers might give you trouble though and decrease your coverage to much. If so, use ball valve timers (Restmo makes a cheap single port digital thats barely ok if you can keep it dry) and problem solved.