One of the best things anyone ever gave me was Radio Shack's best electronics lab kit. 300 projects in one.
I had done several projects but it was without a breadboard. The R.S. kit has a breadboard which is a major convenience.
Also handy is a set or two of 10 jumper wires with small alligator clips.
Cables to connect audio to/from computers, players, amplifiers, etc., with alligator clips.
Spare replacement plugs for audio, power, etc.
Resistors used are mostly in the 100 to 1M range (1/2 W).
In the 1 to 100 range you may find a use for power resistors (1 to 50 W).
Potentiometers, one each: 1k, 2k, 5k, 10k, 20k, 50k, 100k, 200k, 500k, 1M. (Half my projects use one or more pots.)
Capacitors .001 uF through 10 uF. Small values can be ceramic.
Then starting with electrolytics. These get more expensive as the uF value goes up. 10, 22, 33, 47, 100, 220, 330, 470, 1000, 2200, 3300, 4700 uF. 25V rating is adequate for low voltage projects. (Buy inexpensively online from TheCapKing.com)
Transistors, NPN and PNP. Several each small signal (TO-93).
A couple TO-220 size.
If you want to control more than an ampere, consider a few 2N3055 (TO-3).
A couple dozen 1N4001 diodes (1A, 50V). Also get higher ampere/voltage ratings if you think you'll need them.
Assortment of zener diodes. 3 to 15 V.
A grab bag of led's, different colors.
IC's. A few 555 timers (and 556 dual).
3914 dot/bar display driver.
4017 decade counter.
For logic gates you can derive almost anything from NAND gates.
A few 4069 hex inverter IC's.
4066 quad bilateral switch (for analog signals)
If you use 7-segment displays you'll want 4511, 4543, 7447, 7448. Consider an all-in-one multi-digit counter IC.
A few adjustable voltage regulator IC's.
Op amps. Some 741's and 1458 (dual) (Or better types if not too expensive).
324 (quad) can run from single supply.
386 low power audio op amp.
339 comparator.
Fuses. It's hard to choose what is most needed. Get a few of the type needed in your equipment.