Legends of the Streets
Like the name suggests, this is basically the Arcade mode from Initial D: Special Stage, but without the extra courses.
If you still haven't picked any car yet for your garage, make sure to pick one, but make sure that you pick wisely, since the progress is tied to each car, so if you wanna switch to another car, you'll have to do everything from scratch.
How this mode works
Like I just stated two lines of text ago, the first thing to do is to pick your car. Refer to the "What car to choose?" page for more information and some guidance on the matter.
Once your car has been chosen, it's time to actually start the races. If you have never raced before in that mode, you'll be greeted with this screen:

As you can see, you'll only be able to race on the following courses against the following characters:
Myogi
Car
Race conditions
Difficulty (out of 10)
Itsuki Takeuchi
Toyota Corolla Levin SR (AE85)
Day - Dry - Counterclockwise
Toyota Corolla Levin SR (AE85)
Day - Dry - Clockwise
Shingo Shouji
Honda Civic SiR-II (EG6)
Night - Dry - Counterclockwise
Takeshi Nakazato
Nissan Skyline GT-R V-Spec II (BNR32)
Night - Dry - Clockwise
Usui
Car
Race conditions
Difficulty (out of 10)
Two Guys from Tokyo
Nissan Silvia Spec-R (S15) Day - Dry - Counterclockwise
2
Nobuhiko Akiyama
Toyota Altezza RS200 Z Edition (SXE10)
Day - Dry - Clockwise2
Mako Sato and Sayuki
Nissan SilEighty (RPS13)
Night - Dry - Counterclockwise6 Sakamoto
Suzuki Cappuccino (EA11R) Night - Wet - Clockwise
7
Akagi
Car
Race conditions
Difficulty (out of 10)
Kyoko Iwase
Mazda ɛ̃fini RX-7 Type R (FD3S)
Day - Dry - Uphill3
Kenta Nakamura
Nissan Silvia Q's (S14)
Day - Wet - Downhill4
Keisuke Takahashi
Mazda ɛ̃fini RX-7 Type R (FD3S) Night - Dry - Uphill
7
Ryosuke Takahashi
Mazda SAVANNA RX-7 ∞ III (FC3S)
Night - Dry - Downhill8
Akina
Car
Race conditions
Difficulty (out of 10)
Kenji
Nissan 180SX Type X (RPS13)
Day - Dry - Uphill2
Koichiro Iketani
Nissan Silvia K's (S13)
Day - Dry - Downhill3
Wataru Akiyama
Toyota Corolla Levin GT-APEX (AE86)
Night - Dry - Uphill8
Takumi Fujiwara
Toyota Sprinter Trueno 3door GT-APEX (AE86)
Night - Dry - Downhill9
Happogahara
Car
Race conditions
Difficulty (out of 10)
Toru Suetsugu
Mazda Eunos Roadster (NA6CE)
Day - Dry - Outbound4
Daiki Ninomiya
Honda Civic Type R (EK9)
Day - Dry - Inbound8
Smiley Sakai
Honda Integra Type R (DC2)
Night - Dry - Outbound8
Tomoyuki Tachi
Honda Civic Type R (EK9)
Night - Dry - Inbound9
Irohazaka
Car
Race conditions
Difficulty (out of 10)
Seiji Iwaki
Mitsubishi Lancer Evolution IV RS (CN9A)
Day - Dry - Downhill6
Kyoichi Sudo
Mitsubishi Lancer Evolution III GSR (CE9A)
Day - Dry - Downhill7
Kai Kogashiwa
Toyota MR2 G-Limited (SW20)
Night - Dry - Downhill8
Keisuke Takahashi
Mazda ɛ̃fini RX-7 Type R (FD3S) Night - Dry - Downhill
9
In order to progress, you simply must beat these guys, and once you do it, you'll see that their character portrait image has a single red slash above it:

What this means, is that you have beaten that character one time, but what happens if you try to beat him again? Well, check these cutscenes, now they feature rain! Yes, as you might have seen already in the rival selection screen, you are doing the exact same race but now, instead of doing them in dry conditions, now they must be done with rain (with the exceptions of Kenta and Sakamoto since those always feature wet conditions).
Beat them once again, and their portraits will now display a double slash on their portrait, indicating that you have succesfully now beaten them twice.

Walkthrough of this mode
First of all, defeat all of the characters listed above at least once in order to unlock the following secret race:
Irohazaka
Car
Race conditions
Difficulty (out of 10)
Takumi Fujiwara
Toyota Sprinter Trueno 3door GT-APEX (AE86)
Night - Dry - Downhill9
Beat Takumi at least once, and you'll finally unlock the final race, which is this one:
Akina
Car
Race conditions
Difficulty (out of 10)
Bunta Fujiwara
Subaru Impreza WRX STi Coupe Type R Version V (GC8F)
Night - Dry - Downhill10
Manage to complete this challenge and you'll be greeted with the credits, in which you'll see a replay of your race against Bunta.
Post-credits content
Once you have seen the credits, you'll be greeted with a Game Over screen. But, this doesn't mean that the game is completed (with the car that you have chosen). Like we said before in the "How this mode works" section, you can play against everybody in the wet condition, so that'll be your main objective for now.
This time, you can do every race in any order you may wish, but that doesn't mean that you can simply beat Bunta again and call it a day, no, you must win against everybody in order to actually beat this mode and be greeted again with the credits.
Once that you have fully completed the Legends of the Street mode, if you try to enter again the course selection with the same car, you'll realize that your progress has been wiped entirely (except the fact that you can race Takumi in Irohazaka and Bunta from the get-go), but that's not entirely true. The game doesn't tell you about this, but the game has increased a notch (we'll call it NG+1/New Game Plus 1 from now on) its difficulty (not to be confused with the difficulty setting in the options menu, since that difficulty modifier stacks up alongside the one you have in this mode).
(star photo with difficulty)
While at first glance you might don't see too much of a difference, the more times you complete this game mode with the same car, the higher the difficulty will be, so as you can see, this game mode is designed to be completed multiple times instead of just one in order to enjoy it at its fullest potential. So for example, if you have managed to beat everybody twice seven times, you'll be playing in NG+7/New Game Plus 7, and unless your car is a meta car and your times are relatively close to the world records, just forget about giving it a chance.