The Chiliad Challenge is only available after you get your Cycling skill up a bit, so if you're not eligible to participate, try doing the Courier challenge in Los Santos beforehand. The Challenge opens up between 7:00 and 18:00; to start it, fly or drive up to the very top of Mount Chiliad and hop atop the Mountain Bike there.
Chiliad consists of three different races, all downhill races on the mountain bike. You'll obviously want to have as much Cycling skill as possible here, as the mountain paths will have plenty of opportunities for falling. The annoying thing here, though, is the fact that you don't get reset to the top of the mountain if you lose or get disqualified, meaning that you'll have to rescale the mountain after each unsuccessful attempt to try again. You'll also automatically get changed over to biking shorts when you try one of these; you'll have to return to a wardrobe to change back into whatever you were wearing before.
Beyond that, though, these are straightforward races. Get to the finish line first!
The BMX challenge is located in Glen Park in Los Santos; you probably noticed a bike park, complete with half-pipes, when you passed through during the bike-back-to-the-Grove mission early in the game. When you head back that way, then get on the BMX bike near one of the half-pipes, you'll start the BMX challenge. You'll need to have upgraded your Cycling stat before you can attempt this challenge, though. (If you aren't eligible, you can probably get the skills you need by running the Courier mission from Roboi's Food Mart in the Commerce district.)
In this challenge, you'll have to get through a number of coronas scattered around the park; some are scattered about on the ground, but many of them will require you to catch air before you can get them, meaning that you'll need to bunny hop. Although you don't have to max out your cycling stat to start the mission, you may need to get fairly close in order to actually complete it, as the highest rings require some large bunny hops. You're also on a timer for this challenge; you start with only ten seconds, but each ring you get adds ten seconds to your time. Go for the easy rings first, then use the buffer time you build up to focus on the more challenging ones.
If you fail to get all of the rings before the timer runs out completely, you can start it again by riding away from the park with your bike, then riding back in. This will cause the bike to respawn, allowing you to jump back on for another try.
In order to unlock this mission, you'll have to bypass the Cesar Vialpando mission given to you by Sweet early in the game. After that's done, you'll notice a man standing outside Unity Station in Los Santos; if you speak to him, he'll offer to let you into the lowrider competition. In order to compete, you'll need a vehicle with hydraulics, so if you don't have one already, hit the chop shop and order up a set.
After you begin, you should recall what to do from the story mission; watch the arrows as they float across the screen, and press the right analog stick in the corresponding direction when it overlaps with the circle. You only need to win this once (not including for the Cesar mission) to get your credit towards 100% game completion.
The NRG-500 Challenge is identical to the BMX challenge, save for the fact that it takes place on a superbike, and is located at the San Fierro docks. You're faced off with another assortment of red coronas, which are situated around what's essentially an empty, smooth-sloped pool, some of which are easy to hit, but some of which are quite difficult to get.
First off, grab the easy ones, on the bottom of the pool and around the upper edge, to build up your time reserves for the more difficult jumps. Getting the highest coronas will require you to build up a lot of speed, which can be difficult to do, given the relatively small size of the pool; those aligned on either side of the long axis of the pool will be maddeningly difficult to hit properly, since you'll have to jet down the opposite side to get enough speed to make the jump. Luckily, the highest coronas leave small shadows on the upper edges of the pool, so aim for these. You may need to shift your weight backwards on the bike before you can successfully hit the high coronas; otherwise, you may sail forward when you jump.
This will probably be a bit more difficult to complete than the BMX challenge for most people; the best thing about it is that you can just ignore the competition and race away in the NRG if you're just looking for a fast bike to tool around in.