I first would like to give full credit to the others who posted solutions. I am just giving additional insight or tips from my own gameplay experiences while playing this game. There may be some challenges that will have similiar solutions to what has already been said.
Note: When you first start the game, you will only have 30 challenges, as you progress the story you will unlock more.
Nut Shots (25 times) - Remember this is for shooting enemies in the nuts, not punching
Grenade Kills (50 times) - I found this best to do when taking down gang operations on the map. Many enemies will be standing next to each other. You can also upgrade your grenades at Friendly Fire to do more damage and have a bigger blast radius.
Cars Destroyed (300 Cars) - This should happen naturally as you play the game and do survival missions. If you are short I would recommend using a rocket launcher with infinite ammo to finish up. Upgrades available at Friendly Fire.
Helicopters Destroyed (60 Helicopters) - Most of these can be done by playing the campaign and survival missions. You can always get a high police notoriety and they will send choppers after you. You can upgrade the rocket launcher to lock on the targets to heat seek them out as well.
Boats Destroyed (25 Boats) - As stated in a previous guide, the "Ho Boat" mission will rack up a few boats destoyed. You can also get some police notoriety and ride around in a boat and police boats will chase you that can be destroyed.
Tanks Destoyed (25 Tanks) - This can be done in survival and mayhem missions. As always, if you have a high enough police notoriety they will come after you with tanks that you can destroy.
Carjackings (250 times) - This will come naturally while playing the game. If you are short, using the LB + Y (quick car jack) will make it go faster.
Mascots Killed (200 Mascots) - Easiest to do in Professor Genki's gameshow because only mascots are enemies. You can always replay these missions to rack up the kills you need.
Hoods Explored (18 Hoods) - This should happen automatically as you play the game.
Shortcuts Found (50 Shortcuts) - There are actually 263 shortcuts in the game, you just need to find 50 of them. Just look for driveways between buildings and alleys and you should find 50 in no time.
Hostages Taken (50 Hostages) - Stand in an intersection and steal cars that have passengers in them. The game will prompt you to hit Y to take hostages. Hit the Y button to start the hostage activity, then stop the let the hostage out. You do not need to complete the activity for the hostage to count.
Stores Robbed (15 Stores) - There are 34 stores, you just need to rob 15 of them. Enter the store, hold LT and point your weapon at the clerk, continue to hold the reticle on them for a few seconds until you are successful.
Streaking Time (600 seconds) - Go into your wardrobe at your crib and choose to remove all clothing. When on the street, hit down on the d-pad to start streaking. Get close to as many people as you can to keep the activity going and rack up more time. You can still use LB to run which helps keep it going.
Big Air Distance (5,000 feet) - There are many jumps around the city to use. If you are at the airport, next to the runways there are triangular objects that stick out of the ground you can use as jumps. This is nice because you can also use the runways to get some good speed first.
Gang Members Taunted (50 Members) - Every time I saw a gang member walking on the street I would get out of my car, walk up next to them, and hit left on the d-pad.
Vehicle Surfing Time (600 Seconds) - This can be done alone but the computer drivers act different when you are surfing and drive sporadically and turn around for no reason. This is most easily done with a friend in co-op. Just jump on the hood of a car, wait for it to start moving and hit Y to start the activity.
Windshiled Cannon Distance (1000 feet) - Get going fast enough and ram into something and you will be ejected from the car through the windshield. I had success using the Emu with nitrious equipped.
People Run Over (500 People) - Just drive on the sidewalk and hit people. This will actually most likely be done naturally as you play the game.
Oncoming Lane Distance (100,000 feet) - Just drive in the oncoming lane to get this. Easiest done on highways.
Wheelie Distance (10,000 feet) - Get a motorcycle to do this. I did it at the airport on the runways so there is no other traffic to get in the way.
Powerslide Time (300 seconds) - Hold A and crank the wheel of your car while excelerating to powerslide. This is best done at the airport so you have more space. I had success using the car named Torch.
Decker Specialist Killed (50 Specialists) - This can be done during survival missions in that gangs hood. You can also get a high enough gang notoriety and for this particular gang and they will bring out specialists that you can kill.
Luchadore Specialists Killed (50 Specialists) - This can be done during survival missions in that gangs hood. You can also get a high enough gang notoriety and for this particular gang and they will bring out specialists that you can kill.
Morning Star Specialist Killed (50 Specialists) - This can be done during survival missions in that gangs hood. You can also get a high enough gang notoriety and for this particular gang and they will bring out specialists that you can kill.
Near Misses (500 of them) - Just drive close to another car without hitting it. Most quickly obtained in oncoming traffic.
Car Torpedo Damage (2000 Damage) - Get your vehicle going fast and hit Y to bail out, whatever your car damages after you bail will count. I would send a powerful vehicle like a SWAT vehicle into oncoming traffic and it didn't take long.
Vehicle Mod Spending (20,000 dollars) - Just go to Rim Jobs and spend 20 grand on upgrades and modifications.
Human Shields Used (150 shields) - Run up to enemies and hit Y to take them as a human shield.
Two-Wheel Distance (2500 feet) - Best done on the highway. Veer over to the median and get your front-right and back-right tires up on median and drive. This will leave only two tires on the ground and will be done easily.
Multi-Kills (500 Times) - Best used with grenades or rocket launcher. You can rack up gang notoriety and shoot the cars with a rocket launcher as they pull up and get multi-kills everytime.
Basejumps Nailed (15 times) - Jump off the helipad at the Saints HQ and hit Y to deploy parachute. Then hit Y again to create a basejumping target to land in. Land in the target, run back to the elevator to the top of the Saints HQ and repeat.
No-Weapons Kills (25 of them) - Hold B for your weapon wheel and choose the top-middle option for no weapon. Then kill people with LT and RT melee attacks.
STAG Commandos Killed (25 of them) - This can be done in later story missions in the game easily. You can also rack up a police notoriety later in the game on they will attack you with commandos.
Stoppie Distance (1000 feet) - Use a motorcycle, hit LT and push up on the left stick to stop. This is done the easist at the airport on the runways.
Compliments Paid (50 of them) - Hit right on the d-pad near someone to pay them a compliment. This can be done to anyone, friend or foe. Just find people walking on the street and compliment them.
Distance Traveled (2500000 feet) - You will get this naturally by playing the game. If you are short flying a plain is a fast way to travel.
Gang Ops Defeated (32 of them) - I suggest doing these last. First buy all the properties and shops, then do all the activities. When you look at your map the sections of the map that you completely control will turn purple. The sections that aren't purple you know there's a gang operation in that area. Most of the gang ops you will find naturally by playing the game, but some are tricky to find. If you are having trouble, take a helicopter and check on top of buildings and skyscrapers. Flying around in a chopper is also an easy way to spot gang ops on the ground. Just look for a group of red dots and get close to them.
Survial Completed (26 of them) - These are done by answering the phone by hitting Y when your phone rings. Only 25 of the 26 survival activities are available to you at the beginning of the game, so if you aren't getting a call for the 26th survival don't worry. The 26th survival activity doesn't unlock until sometime in Act 3.
Emu's Destroyed (50 of them) - Find an Emu, put it in your garage, retrieve the car, then blow it up. Go back to the gargage, retrieve the Emu and blow it up. Repeat.
Kill Brutes (50 of them) - This should pretty much happen naturally as you play the game. If you are short rack up a high gang notoriety and they will bring brutes in the back of pickup trucks to attack you.
Headshots (500 of them) - This should easily happen if you concentrate on headshots from the beginning of the game. I found it pretty easy when just using the pistol. There is plenty of killing to do so this won't be difficult.