one thing you need to understand about gears is that each gear has a speed range
1st 0-30km
2nd 20-60km
3rd 50-90km
4th 80-130km
5th 120-180km
this is hypothetical since this changes from car to car. ie a peugeot 205 may be 20-60km range in 2nd gear whereas a ferrari could be 40-90km. you will need to familairaize yourself with the car you are using to understand what the power bands are.
in racing games change gear just before you reach it's limit. ie when the rpm needle gets into the red. some cars will allow you to go a few hundred rpm more. anyway you will hear the engine make a distinct noise at it is at it's max. the idea is to gear up just before that point.
for braking you want to reduce gears gradually to make the most of engine brake to help better breaking. if you downshift too quickly you will hear the gears make a high pitch noise. this is not ideal and in real life, you would damage the transmission. ie dont go into 1st if you are cornering at 60km, you should be in 2nd or 3rd depending on the car.
when you get familiar yourself with this then you can tune and change the gear ratios. ie make them longer so you reach higher top speed (drawback is less acceleartion, but important on tracks with long straights ie Indy 500)) or lower the gear ratio where you never reach that top speed on a track and never get to use 7th gear (ie Monaco)
dont buy a wheel just yet, learn to do this with controller. it's easy. can't understand those that say it's too hard for a controller? not that many buttons to use in racing games (most important are accelerate, brake, steer, gear up and down - unless one feels that windshield wipers, horns, and headlights are used a lot

)
anyway practice, practice and you will get better - and you will crush the times with an auto tranny.