I have wanted to get into the Forza series for a while, but the last one I played was 3. I heard they are pretty good, especially the Horizon series, but who knows.
I've heard of the forza series on Xbox, though I never played it I over heard that Forza 2 is when that series started to shine
as for Gran Turismo stay away from "Gran Turismo 3 A Spec" that game is pointless to work for since there is not many cars to play with. I own the first 4 Gran Turismo games and I don't like "Gran Turismo 3 A Spec" much at all. if you feel brave enough you can buy Gran Turismo 2 for PlayStation 1 for cheap and it is well worth the puchase it is cheap as of (1-17-2018)
https://www.amazon.com/gp/offer-listing/B000034DC7/ref=sr_1_1_olp?ie=UTF8&qid=1516224859&sr=8-1&keywords=gran+turismo+2+ps1best played on PlayStation 3 where you can create a built in memory card and have the maximum number of cars while going on a budget. The PS1 memory cards are expensive and are a waste of money in my case.
PS all gran turismo games have both arcade mode and simulation game play at least 1,2 and 4. but Gran turismo 2 is suppose to come with 2 cds one for arcade and one for simulation.
Gran Turismo 1 comes with only 1 disc