Get a custom helmet

Figured it out. Part of the problem was that the server situation on day one was preventing texts and calls from Simeon from triggering.

As you level he sends one telling you about purchasing a garage for your car. This happens at around levels 5-6. You can find all the properties by going to the in-game website: dynasty8realestate.com. The properties will also show up on the map if you drive close to them. They have a symbol that looks like a house; it's also very similar to the convince store icon.

By level 10 it looks like most of the available content has been opened up.

Heist are currently not available, and will arrive in a later update.

In single player the parachute unlocks in ammunation after you complete the mission minor turbulence.

In GTA Online the parachute (and the parachuting activity) unlocks at level 11. Then it becomes available for purchase at Ammunation.

You can also find parachutes around the map, such as on top of the FIB building. According to Gizmo3k you may pick these up and use them before unlocking them at level 11. Another location is by the cargo ship on top of the loaders and on the large train bridge over the river leading to the Alamo Sea. See here for more detailed information about unlocks in GTA Online.