Well, within Campaign you have:
-GT League which is equivalent to GT Mode... so an offline set of races against AI. Different criteria for each race.
-Driving School is a set of short tests, introducing you to the car handling and control
-Circuit Experience contains all of the (original) circuits included in the game, broken down into sectors with awards for time achieved. These are good for learning the tracks
-Mission Challenges are a variety of tasks, again with awards for their achievement. Some are overtaking challenges, some are top speed challenges, some are endurance challenges
Then there is Sport Mode. Sport Mode consists of:
-Daily Races. These are 3 races which update daily. Race A is usually a 'One Make' race with a car provided. Usually these are slower cars and shorter races (often 3 laps). Race B is then a medium length race (4 or 5 laps) usually in a Gr. 4 or Gr. 3 car of your choice and from your garage, although this is sometimes also a One Make race (and twice this week has been Gr. 1). Race C is a long race, between 8 and 12 laps (depending on the length of the circuit) in a Gr. 1, Gr. 3 or Gr. 4 car of your choice, from your garage. You are able to set qualifying times for each of these races before entering, the qualifying time sorts the grid when you enter the race.
-FIA Races. There are two types; Manufacturer series and Nations Series. The Manufacturer series is run over a period of days with a different race each day, your choice of car is fixed before your first entry by signing a sponsorship with a particular manufacturer - you will then use this manufacturer's Gr. 3 or Gr. 4 car for the entirety of the series. Points are awarded per competition and the winner is the person with the most points at the end of the series. Entries are hourly and include closed qualifying (so a full circuit including those entered into the same race as you) and a long race. The Nations cup is similar but (I believe) there are some events which are 'One Make' (vehicle provided) and others you have a choice of Gr. 3 or Gr. 4 cars from your garage.
Edit:
Arcade is arcade, pick a track, pick a car and race the AI
Lobbies are race rooms with user-defined restrictions. These do not count towards your GT ratings.
FIA Races: you can only enter once per day. Entry periods 1 hour apart and there are 4 (?) per day.
On the Daily Races:
Race A is a grid of 12
Race B is a grid of 16
Race C is a grid of 20
Your qualifying time stays for the remainder of the day, so you can enter race after race without re-qualifying. The races cycle every 15, 20 and 30 minutes (IIRC). You can often enter Race B, complete the race and be back in the home page in time to enter it again.
In all Daily Races and FIA events, your DR (Driver Rating) and SR (Sportsmanship Rating) are taken into account and can increase or decrease depending on performance. Your current DR and SR are top right on the homepage - DR is a letter (from E up to S) in a circle and SR (again, from E to S) is a letter above two hands shaking. Very simply: Winning increases DR, losing decreases it. Not crashing increases SR whereas crashing or cutting corners, decreases it
Edit edit: personally I use sticks to steer and triggers to accelerate and brake, but that's a personal preference...