To rack up the coins easily, pick one of the easiest missions, which is the fast highway mission. All that you have to do is accelerate, weave around cars, and then cross the finish line. If you pick the normal fast mission, then your goal is to do the exact same thing, except you have turns to deal with too. Just make the turns like normal without running into any cars.
The drift stage can be the quickest one to beat if you get used to it. One easy way to rack up the drift time bar is to start driving in the middle of the first intersection, and turn it into a donut. Crank the wheel as far to the side as you can and then hit the accelerator. Spin a few times and then continue down the road you’re supposed to go on (if you stay in the intersection too long, then other cars will hit you).
The fuel efficiency and VIP challenges are completely different. Drive slow in these ones, tailgate other cars, and let off the gas as often as possible in order to save gas. To increase the level of comfort, keep at a consistent speed.
Do these often enough to earn silver coins. To earn free gold coins, link the game up to Facebook. Complete various achievements in order to earn free silver coins (some of the ones on Game Center will earn you silver coins for free but not all of them) or simply beat missions over and over (as opposed to losing missions).
Go to the car menu and you can not just shop for new cars (which are expensive and cost gold coins) but upgrade various parts of your current car, which is far cheaper and only costs silver coins. Upgrade comfort in order to complete the VIP mission more easily, and upgrade durability in order to withstand more bumps.