There are challenges for every activity and diversion in the game. As well as each weapon having a challenge, each power, and killing certain kinds of enemies an arbitrary number of times. Not all of the challenges are available right away, you will need to obtain the related item for it's specific challenge to show up. There are 102 challenges in the game to complete.
Challenge Milestone Challenges:
These are challenges for completing a certain number of challenges. There are challenges for completing 1, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 101 challenges. You will complete these as you complete everything else.
As outlined in the "Plague of Gat" achievement. In the bottom left part of the map there is a building that is guarded by demons. There is a vertical street that divides Azazels Court and Seers Tongue, that is where this building is. Go to this building and start building notoriety, this building has fast and infinite respawns. For the sake of quickness and spawn rates, makes sure not to kill the demons that float around in a bubble. They summon more enemies. You can farm all kill types for all achievements and challenges this way.
This is a little different than straight up grinding weapon/power kills. For certain types of demons you have to go to certain districts, and for others you need a certain notoriety level.
Legionnaires spawn on notoriety bar 4/5. Once you are on the appropriate notoriety level, run around a little bit, and the Legionnaires will eventually come to you. Usually in ones or twos. 2 also spawn with every 7th Dex that spawns.
Tragedy Demons only spawn in the top right area, the Den, so if you need Tragedy demons head there and start getting your notoriety up. The Tragedy Demons have Theater Mask faces. In the same vein Bro Demons only spawn in Shanty town and dress in varying colored polos with shorts. These guys usually show up in Monster Trucks.
Tanks spawn at higher notoriety levels. Grenadiers and Enchanters spawn as your notoriety goes up. Arch-dukes spawn at Marshalling grounds when you're trying to take them over and most of the time when you reach notoriety level 6. The other times Genki will spawn and it is a challenges to kill him as well. He also gives you the Wrath Seven Deadly Sins weapon when you kill him.
You will need to drive a comet around while Balls of Fire is initiated for a certain amount of time. If you hit something while BoF is activated the comet will automatically blow up. So you will need to weave through traffic to get a good run. These spawn everywhere. I would recommend snitching one on the highway and trying to drive it on there. There is also a parking lot near Snowball's Chance Casino that spawns one.
In the same parking lot is a Rambulance, get in and start the associated diversion. Then just start running over Husks. You will now need to steal a Monster Truck, these start spawning by the numbers when you have notoriety. Once you have it start the diversion and start running over other vehicles. Be mindful of doing this on bridges or near edges as you might lose control while running over vehicles. You will also need to steal sinterpol armored vehicles. To do this you must damage but not destroy the vehicles and then hop in first chance. You will then need to deliver it to a designated waypoint/location.
Barnstorming is when you fly through gaps in buildings, arches, and in between certain buildings. You will get a good number of these doing the Hell Blazing activities as well as flying around Downtown/Shantytown. Everything else can be covered by saying just do your Pledge Rushes, kill the spires, survive the survivals, and complete the altars.
So we have 4 kinds collectibles here, Soul Clusters (for upgrades), Tomes (for character commentary), Glyphs (to get the Boom Chicka), and Commentary on hell (From Gat and Kinzie). Aside from what you need/want for soul clusters for the sake of upgrading I would not go out of your way to get collectibles until you have the Collectibles finder which you can buy when you reach level 15. This allows you to see almost exactly where every collectible is and enables you to set custom waypoints on them. Also for soul clusters remember to buy the ones from the upgrades menu or you will come up short.
There are 3 activities, Hell Blazing, Salvation and Torment fraud. Each one is unique in it's requirements and playstyles. Firstly Hellblazing is racing activity. You are flying around hitting checkpoints (can't skip any) and trying to finish as fast as possible. Then there is Salvation, in which you're attempting to catch souls/husks before falling to hell or ascending to heaven. This is more free flight instead of a chosen path. If you're struggling with either of these upgrade your flying abilities so you can stay airborne longer and get more flaps. Also upgrading your stamina will also help.
Lastly there is Torment Fraud, this is the hell version of insurance fraud. You run around as a husk of a person that the Saints knew on earth trying to shave off time in hell by getting yourself hurt by throwing yourself in front of cars, broadsiding buildings and slamming into the ground. Once you get some momentum the gold requirements for this are very simple. The real trouble comes from the "Lets Bounce achievement" not the gold requirements.
There are challenges for clearing out each district which will come with clearing out the other challenges. You will also need to drive any vehicle around for a little bit. The kicker is fully upgrading all 20 weapons which costs $2,216,000. So depending on how many upgrades you bought might need to grind out some money like I did. So try to only buy the minimal amount you absolutely need to get through the game.