Getting Doc’s Notebook
When the game begins, you'll take control of Marty in a dream sequence. Go through the beginning conversation and move over to the toolbox (it's red) which can be found to the left of the screen by the van and left click it. Another conversation will play out and after it, Marty will wake up.
When you next gain control of Marty, you'll be in a room with Marty's dad and also Biff Tannen. Feel free to talk to the characters here, but what you want to do is click on the Town Square model, where Biff is. Select any conversation option and then move to the left, where you can find a guitar by Marty's dad. Click it. Then, click the amplifier controls nearby.
Open your inventory, select the guitar and use it on the amplifier then. Talk to Marty's dad and choose the 'I can fight my own battles' option then and exit the conversation, then click the guitar. This'll allow you to get Doc's notebook.
Going Back in Time
When you're outside with the DeLorean, click on the car. From here, take the shoe and then click on the tape recorder. Click on the exit button from here and use the shoe on Einstein. When you gain control of Marty again, click on the buzzer. Choose the 'I've got something to show you' conversation option when prompted and use the shoe on Edna. You'll go up to her apartment then.
Once there, click her and choose the newspaper conversation option, followed by the shoe one. Choose the newspaper option next, followed by the speakeasy one. Next, the 'where was that speakeasy?' option. Exit the conversation then and use the binoculars. Now click on the radiator. Click on the newspapers then and you'll be done here.
Next, choose any dialog option when Marty's dad talks to him, then click the time circuit switch followed by the display.
Finding Young Doc
When Marty gets approached by a woman, choose any options you like in the conversation. After that, head straight to the jail, unless you want to explore. In the conversation with Doc, choose the 'back into the past' option and then the 'break out of jail' one, followed by 'where can I find your younger self?'
After that, go inside the Soup Kitchen which is to the right of the jail/station. After the cutscene, move to the telephone and click it. Exit then and go to the courthouse. Young Doc will come out after you click on the doors. Follow him and click to initiate a conversation. Choose the rocket drill option, then the 'I've got something important for you to do' option. Following that, choose the rocket drill option again. Next, go into your inventory and select the tape recorder, then use it on Young Doc.
Go back to the jail and click on the window. Select the 'I recorded your muttering option'. Now go back and talk to Young Doc. Tell him the answer you got from Doc.
Getting the Rocket Fuel
When you regain control after the cutscene, go and talk to Kid Tannen who's getting his shoes shined. When prompted, choose the 'shine your shoes' option. Ask about the racket, then about it again, before choosing the Arthur McFly option. Then ask about the hat. Ask about having some peanuts, then choose the 'what the hell is that?' option.
Go back to the Soup Kitchen after the cutscene. Talk to Young Doc and ask about where you can get some alcohol. Then, talk to the guy behind the counter and ask if you can try some soup. Choose any option after that. Then go around and try the door that's barred. You won't be able to open it, so talk to Young Doc and see if he has any ideas.
After that, keep trying the soup and going around once the guy behind the counter moves to add spice to the soup, until Young Doc has an idea. When he has the idea, hit the pipe and Edna will come in.
Go outside and talk to Edna. Ask about the scoop, then other story, sober society and deliveries. Say you can help deliver soup to the society. Then say they can meet at the Brown Residence. Say that the society needs a delivery then.
Young Doc won't be happy, so go through the dialog options and threaten him.
Finding Arthur and Preparing the Rocket Drill
After the conversation with Young Doc, use the hat on Einstein. Click the buzzer. Then click Arthur and choose any dialog, before returning to where the courthouse/jail/soup kitchen is. Go to the barber shop. After the cutscene, use the recorder on Kid Tannen. Click Einstein. Then go back to where Arthur is and use the recorder on him after pressing the buzzer. Use the subpeona on him then.
Once you're in the lab, choose any dialog option. Now then, this puzzle is different for everyone, so you'll need to pay attention to the different things Young Doc says, especially the capitalized words. The capitalized words help indicate which apparatus you should use. When done, choose any answer to give to Young Doc about how long it'll take to get a response.
Preparing the Rocket Drill and Rescuing Doc
Pick up the drill remains when you regain control, then click Edna's bicycle. After the next scene, move to the right and click on the antenna, followed by the toolbox. Now move to the left of the screen and use the tire iron on the wheel. Go to the driver's window from here and click it, then use the antenna on the keys.
Talk to Doc and choose the distraction option. Then use the antenna on the keys. Now go back around and use the keys on the padlock. From here, use the item you got from the tire on the gun and then click on the gun to collect it. Use the gun on the shackles and then select the throttle option after clicking on Doc.
All screenshots and references from Back to the Future: It's About Time!.