I played one of the earlier Grids, it didn't really impress me that much. I enjoyed Need for Speed Shift slightly more.
As for Forza vs Gran Turismo...it depends.
I have Gran Turismo 5; my roommate has Forza 3.
GT5 has a greater variety of cars(compared to the later Forzas as well), but some car lack interiors. The interiors are just plain black for lesser known vehicles. GT5's car sounds also sound more "arcadish". You'd drive an A class car and it would sound similar to a C class car. I'm not sure if this is a thing in Forza, but you have to do a lot of licenses in GT5, which gets tiresome because they're basically just normal challenges that don't really indicate whether you have actual racing skills.
Forza has rewind! If you crash, oversteer/understeer you can rewind and do a section again. As far as I've seen in demos and reviews, Forza also has greater car customization than GT5. GT5 basically lets you change colors(you also have to buy "special" colors with real money), but Forza lets you put customizable logos and banners on your car.
As for actual gameplay GT5 is much more of an arcade game compared to Forza 3 and 4. In Forza 3 and 4 damage is a serious element. If you crash, your car's performance decreases, the damage is clearly visible and can be felt. GT plays like bumper cars. You'll see damage in forms of scratches or slight displacement, but it won't really decrease your car's performance. I really don't know about Forza Horizon and onward, but GT feels a lot easier compared to Forza. It requires less knowledge of technical aspects of vehicles and vehicle physics. In Forza 3 and 4 it's advisable to do calibration to faster vehicles as it's normally harder to break and steer at the same time.
This is just my personal opinion, but I would go for an Xbox and Forza next(If I wanted a racing game). GT has stayed in its original class as an arcade game, but Forza is mechanically and technically progressing.
“Conquer yourself rather than the world.”
― René Descartes