The Abstergo challenges are just like the guild challenges from previous games. I will list a few from each category. If you negative vote this please tell me how I can improve it.
1. Perform 10 assassinations. You can get this in Havana, run around and kill guards without triggering conflict.
2. Synchronize 5 veiwpoints. You should get this while in Havana, there are 2-3 points in the island you chase Duncan on.
3. Kill 10 enemies from hiding places. This can be done in Havana. Hide in a crop field and whistle to get an enemy's attention.
4. Perform 10 air assassinations. This can be done in Havana. Climb up to a roof top and assassinate from above.
5. Perform 5 double assassinations. This can be done in Havana. Run up to two enemies with your weapons drawn and assassinate, you should stab both enemies from behind.
6. Complete 3 assassin contracts. There are 6 to complete in Havana.
7. Drink 5 glasses of rum. This can be done in Havana. Interact with a bottle of rum on the bar counter and drink 5 times. You should also get the Hungover achievement.
8. Capture 5 forts. If you follow my guide for the King Of The Castle achievement you should get this pretty easily.
9. Capture 3 ships. Board a ship, kill the crew, send to your fleet.
10. Reset your wanted level by bribing a local officer. I got this after I captured my first fort, there is always an officer near the dock. I believe there is a mission that takes you to an island and shows you what an officer does.
11. Collect 10 Animus fragments. This can be completed in Havana. There are 15 fragments to find there.
12. Collect 5 song sheets. These are the music notes you see on the map, Havana has 7. The first island you are on has 1. You should get this before leaving Havana.
13. Disarm 5 enemies. Make sure you have fists selected, counter an attack, press A button to grab their weapon, doesn't matter if you kill them. You can get this is Havana.
14. Achieve a 3-enemy kill streak. This can be done in Havana. If you have played the previous games you will know how to chain kill. Chain kill 3 enemies.
15. Kill 5 enemies with a headshot. This can be done when you get a pistol. Aim at their face and fire.
16. Skin 5 animals. This is easier to do when you get the pistol. Kill an animal and skin it.
17. Kill 2 crocodiles. This can be done in Nassau. Find the swamp to the left of the island, there should be some crocodiles swimming around.
18. Skin 15 animals. IF you do a lot of hunting before continuing the story you should get this pretty early. I recommend the crocodiles in Nassau, kill a few, run towards the middle of the island, come back. They should respawn.
19. Loot 10 bodies. This can be done in Havana. Kill 10 guards, loot the corpses.
20. Hire 5 groups of dancers. This can be done in Havana. Find a group of dancers and hire them.
21. Possess 5000 R. This shouldn't be too hard if you plunder ships regularly. Sell the sugar and rum.
22. Sell a total of 500 cargo to a harbourmaster. I got this by plundering ships and selling the sugar and rum.
23. Pickpocket 200 R. Should be able to get this in Havana. Stand/walk behind an NPC, hold the B button to steal from them.
24. Share 3 social chest locations. You will get this randomly as you play through the game. All I know about it is to be online.
25. Share 3 royal convoy locations. Same as the chests, this will appear randomly as you are online.