For A-spec, I don't upgrade the tires unless the A.I. is on upgraded tires. I try to match them.
For Online, I restrict to sport soft or racing soft depending on the night. The difference between the two is really just a few brake zones. Some tracks let you run WOT on RS tires while on SS tires you have to tap the brakes a few more times. It's really just a matter of adapting and learning the brake zones.
This game has a big following of tuners, sim'ers (simulation fanatics), and gamers (DS3 drivers). Tire selection will reveal the strengths and weaknesses of each driver type.
CS Tires: Good for sim'ers, so-so for tuners, and horrible for gamers. Tuners can find some edge on this tire but driver skill can make a bad tune much quicker than a good tune. 4WD cars typically dominate but simulation drivers that are smooth and fast will annihilate the competition. There is typically only one quick driving line so passing can be very difficult. It's more a game of who makes the least mistakes.
SS Tires: Good for sim'ers, great for tuners, so-so for gamers. Sim racers will still enjoy getting the car sideways but properly tuned cars can dominate the field when put in the right hands. The tires are soft enough to require some sort of changes in both LSD and suspension over the stock or default-custom settings. Gamers can keep up but typically make mistakes and whine about falling behind. These tires open up more driving lines and allow for more passing.
RS Tires: Bad for sim'ers, ok for tuners, perfect for gamers. Sim racers typically never enter these rooms because cars will rarely break lose the drive wheels making oversteer virtually non existant unless you really screw up a tune. Tuners can still get an edge over the competition if they can reduce braking distances and squeeze out more G's with camber and other settings. Gamers love the tires because they are more forgiving and allow them to stay up front despite using a controller. The driving line is an open free for all and you'll have to protect the entire track if you want to stay out front. Don't expect to pull away from the pack with your perfect driving line.