You want that icon to be in the green range when the race officially starts. When the race is starting there will be a countdown, and you will be able to rev your engine, and you will see a icon moving along the bar into the different ranges. This bar will have a zone in it that's yellow, green, and red. When the race starts, you're going to see a bar at the top of your screen.
The race also involves manual shifting and timing. Drag races involve tracks that are typically straight and short, so your car will mostly be dedicated to acceleration and top speed over maneuverability and such. Drift races are head to head matches putting you against another car in a round based system where you have to win either best 2 of 3, or 3 of 5. This chapter is going to introduce you into two different racing types, Drag and Drift.
If you would rather watch a video and follow along, WarbyGaming has two videos that can help show you the locations and how to collect them:
There are 100 Gambler's Chips and 30 billboards scattered around the world. You will eventually need to be getting all the Speedruns and Speedtraps, however you won't want to do that until after the game when you can actually complete them easier.
There are several different types of collectibles, but the only ones you should really look at getting right now are Gambler's Chips and Billboards, as they don't require any upgraded cars available later in the game to get. It is wonderful for keeping track of which collectible you've received. PascalMarc (SwissGameGuides) has created a fantastic interactive map which will help you with every collectible in the game, which can be found by clicking here. Doing them now or later is up to you, but if you decide to do them later, come back to this section of the walkthrough. You will also be able to discover all the gas stations and garages along the way, which will help you while you progress through the story as well. You will also get a general feel for the map, which will help you when the races take you to those general areas that you've already spent time in collecting things. When you find these collectibles you'll get money, which will give you a very nice stockpile of cash as you progress through the story, which will help you in purchasing speedcards or new cars to stay ahead of the difficulty curve. It is strongly recommended that if you have the time and the patience, you set aside some time to go out and find them all now before continuing the story, since overall it will help you more.