Similar to you, I bought many components so I wouldn't have to spend much time watching the mailbox. Some of my experience:
While the 1N4148 has its uses, a 1N4007 will work just as well in many circuits (at least simple ones), even if it's not always the best choice. But considering the 1N4148 costs about 2 cents if not less, sure, get some.
Like the other responders here, I see little value in the 1N4001-1N4006. Get the 1N4007, it's often about 2 cents.
Schottky diodes, yes. But skip the 20V ones like the 1N5817. While their forward voltage is extremely low, in my experience they leak some current in the "wrong" direction, even in <10V circuits. So get Schottky diodes that are rated for at least 40V like the 1N5819.
Depending on what you plan to make, some higher current diodes could come in handy as well. As the current goes up the forward voltage will also go up a little, so having some headroom can be beneficial. Other than that, they can be used to protect MOSFETs. Some (many? most?) MOSFETs have an internal diode to protect it, but this internal diode is very small and can be damaged by large currents, turning the internal diode into a conductor and turning the MOSFET into e-waste. A larger external Schottky diode in parallel with the internal diode is recommended in this case.
I actually disagree with the majority here on Zener diodes. I've used them in three scenarios. When I needed a VRM that could handle an input of more than 30V, I used one NPN transistor that was rated for a high voltage (I used an MPSA 42, IIRC), a resistor and a Zener diode. Connect the input to the transistor collector and resistor, the other side of the resistor to the transistor base and run the Zener diode between the transistor base and ground. It's not super precise, but if needed you can use a second VRM (either the same kind or something off-the-shelf) in conjunction with this.
Another scenario is to protect some components. 18V Zener diodes can be used to protect the gate of most N-channel MOSFETs (which is typically rated for 20V max) by connecting it between the gate and drain. Something similar goes for most components with a breakdown voltage. A TVS diode would be better suited for this purpose but may be relatively expensive depending on where you shop.
The third scenario is to simply use Zener diodes as regular diodes. Zener diodes are designed to also function in reverse without instantly breaking whereas other diodes are almost instantly irreversibly damaged when their breakdown voltage is exceeded. In some applications this might be an advantage.
One final note: if you occasionally modify or repair existing devices, consider buying SMD variants of these diodes, either in addition to or instead of through-hole components, depending on your soldering skills. I frequently find myself running out of space when working on some existing device and SMD components can be the answer in that case. They also tend to be cheaper. For example SS14 (similar to 1N5819), SS34 (same but 3A), M7 (similar to 1N4007), LL4148 (similar to 1N4148). For Zener diodes, search for "LL34 Zener". Just beware that LL34/MELF is called a "roll-away package" for a reason and that MELF actually stands for Most End up Lying on the Floor. There are also Zener diodes using other packages (SOT-23 or DO-214 for example) but LL34 is often cheaper and easier to find.