As I understand it, even for the professionals this is a pain, and often constitutes the largest portion of design time. Don't feel bad, this is one of the more difficult things to master. It's further complicated by the speed that ICs are constantly being revised, discontinued, or released for the first time, and that makes it hard to stay current.
Thus the concept of the 'jellybean' is born. When you want to breadboard a circuit to test an idea or slap something together quickly to solve a problem it's a good idea to have some inexpensive standbys handy. These get used when you simply need 'Opamp, Mk. 1, standard grade' and are usually very cheap. The goal is to always have a few handy, so you keep a stock and replenish it like you would a resistor of a commonly used value. This doesn't always work well if what you're doing requires high performance, but if you're hobbying your way along it can be a good way to start.
Engineers (or hobbyists, labs, companies) that have different areas of focus or specialization will have different ideas of what makes a good jellybean opamp, for example, because different component trade-offs are valuable to them. Someone who's doing a bit of everything might have a dozen or more different jellybean opamps to cover their bases.
Being a hobbyist, I personally like this way of doing things as it seems like a fair system for working in the free sample services of the manufacturers. I've got no qualms about asking for a sample of a dual voltage, low dropout, low quiescent current regulator with independent standby and soft power on, but I can't keep a straight face when thinking about asking for samples of an LM317 for 'evaluation' that's been in production for years and years.
(P.S. I just made up that regulator, so nobody ask me where to get one
)
Discrete logic is harder to do this with due to the overwhelming variety of tasks you'd have to cover. Others can likely give good advice on this, I've got no real experience in digital logic or mixed signal parts.
You can also look for circuits online that perform similar tasks to the one you're designing for and see what parts they use. Failing all else, you can always ask for advice here! If you describe what you're trying to do I'm sure there's someone who has the experience to recommend a few part numbers.
Some of my jellybeans:
Opamps for audio = assorted TL07X and TL08X (where x equals 1,2,4 for single, dual, quad), LM833
Opamps general = LM324 quad
Opamps precision = OP07
Comparator general = LM339 quad
3 pin Vreg = LM317T positive and LM339T negative (TO-220), LM317L (TO-92), and a handful of fixed 7805 and 7812
Oscillator = any 555 clone, any 556 clone (dual 555), I'm using NE556
To everyone: what do you guys keep handy at all times?
Hope that helps