You can do this from your crib interface, or from any weapon store. Cribs show up as house icons on the radar, and weapon stores show up like handgun icons.
When you're in a crib or weapon shop, the game will show you which button to press. From there it's just a matter of navigating the menu and selecting a new weapon from the weapon wheel. Each slot on the wheel can only hold one weapon at a time, and the slots are only for specific weapons. For instance, you can't take two pistol-class weapons at once.
While in a crib, you'll see "Press (button) to access crib interface." Pressing that button brings up a menu. Move down to "weapon cache" and select it. Move your weapon wheel to highlight the slot, and select the slot. Then scroll through the available weapons and select the one you want.
From a weapon store, press the button for shopping, and then select "Access Crib Weapons" from the menu. You can then select a slot, and then select a weapon as above.
Some weapons can be purchased, others are gifted to you as a reward from missions, and others you can just pick up off of dead enemies. If you pick up a weapon off a dead enemy, it counts as being "bought" and you can select it from any crib or weapon store in the future.
If you're trying to change weapons at the crib and you don't already own the weapon, you won't be able to. You're limited to selecting from the weapons you already own when at your crib. Interestingly enough, at weapon stores you can switch to other weapons you've already purchased/found without paying for them again.