A thing that one needs to accept (and this can be really hard) is that you simply can't reach perfection, and that there will always be a better solution out there.
As you say, this is a time/experience thing. Of course, it can help to ask around a bit if you are unsure - on forums, IRC, colleagues, etc. As others have suggested, paper catalogues can be great. Another usefull resource can be the product marketing books from IC companies. More than once have I gone "oh, that is a thing?" when scrolling through the analog devices or Linear tech (I guess they are the same thing now) marketing emails.
As for your final problem:
Though I was thinking of keeping a small supply of many others, but I have no idea what the most common are. It sucks when everytime I come up with something I want to build for learning or just something I came up with. I have to order and wait for the components to come in.
This can be hard. There are a few options here:
In general, you can sometimes use "less ideal" components for a first proof-of-concept while you wait for components to arrive: If I need to verify if say a current measurement circuit works, I can just build it up with low-precision opamps and see what gives. This will give me an idea of stuff while I wait for the fancy precision parts to arrive. So some general RIRO opamps can always be usefull, and they don't have to be super-expensive to work in a proof-of-concept.
A second thing is that most places still have some form of physical store you can run to. I have a small supply in my university (though they sell only shitty opamps, no precision or riro ones). Additionally, there is a component store where I can always go check out to see if they have something similar to what I seek in stock. They tend to have most components that aren't to exotic, and again, when prototyping the exact part is generally not vital in first versions. Ofcourse, they are a bit more expensive than ordering in bulk from say RS or Farnell, but I have them when I leave the store, and I don't have to wait.
Another thing to consider is just ordering 5 or 10 of whatever part you order (if it's not to expensive) if you think it might be usefull in the future. I tend to do this with diodes and rectifiers, basic logic stuff, etc.
And a final thing that can be interesting: Look at auction sales of old labs, schools, etc. Most of my parts are from a few bulk purchases when companies or labs closed or such. For just 90 bucks I got myself more than 500 euros worth of precision opamps (and that is just by going through about a fifth of the components I got, still sorting it).