Practice as much as you possibly can. It doesn't really matter if you don't win the race as you're gaining experience with every corner, track and car type. Eventually, muscle memory takes over and everything starts to click into place but you have to put the laps in to get there. There's no trick other than hard work.
If you're struggling for credits, spend an hour or two on quick match. Last place pays more than every career race (up to the pseudo endurance races) and you'll have more cars than you know what to do with. Same with the seasonal races as they're huge payers now. Try and keep your log in bonus topped up to 200% to maximise the credits.
While it's easier to control the car's speed and control with the manual transmission, it's possible to be quick with the auto as well. It sounds like you need to get the basics down first and then think about switching to the manual when you want more control.
If you're making too many mistakes, slow down until you stop making them and then increase your speed when you're more comfortable. If you're struggling on a particular track, watch some replays. If you're struggling with a particular car, try another as there's always an alternative and it may suit you better. Everyone has their own driving style and some cars suit some people more than others.
...and as already said, have a look in the tuning forum and the tune finder to find cars that have had most of the bad handling characteristics neutralised. They'll give you the confidence to push that little bit harder.
Lastly, read the first paragraph again. It's the only way to get quicker.
The best of luck to you.
