The reason I play F1 games is because I want to experience what it would be like to be an F1 driver.
Here are the things that I would like to see to improve the game's realism (order of importance):
Mirrors: In PS3 X360 versions of the game, when driving in cockpit view, the mirrors show a PS1 style graphic of a racing car when it is right behind you. In the PS4 / XO versions of the game, I want to be able to look in my mirrors and see as much detail of what is going on behind me as if I was facing in that direction. Pressing the right thumbstick down is unrealistic and gives you more view than a real F1 driver would have.
Practice sessions: In real-life F1, it is not the driver that decides their own setup and how many laps they should drive in a practice stint. The engineer should give specific instructions as to what they want you to do during a practice session, it's then up to you to carry out the instructions and give them feedback to help them find the right setup for you.
Reconnaissance laps, Formation Lap and Race Start: First of all, you should drive from the pit garage for the race, and either drive straight to the grid or through the pit lane a few times if you want to get an idea of the current track conditions. During the formation lap, you have to drive in a way to get the tyres to the right temperature to get an optimal start. AI drivers should be more dynamic at the race start.
Post-race lap: After doing a race at full distance, it would be nice to have a cool down lap to soak up the result, see glimpses of race highlights on the big screens as we drive past, get some feedback from our engineer / team principle, pick up some marbles, then drive into parc ferme.
Interaction with other F1 drivers: I'm mainly thinking of your teammate and maybe some cut scenes to show how your relationship is developing with them based on your head-to-head results. But, interactions in the green room prior to getting on the podium or with a driver that you ruined a race for (or vice versa) could also be a great addition.
GP2 and GP3: Would love to see this, but would rather time be spent on making the F1 game as great as possible. Could be released as DLC in say Q1 2015?