I'd previously ordered a collection of caps, LEDs, parts to make standalone arduino circuits - 16MHz crystals, 22 pf caps. Have built the first board already from those. 0.1uF (100nF) caps for noise decoupling.
Maybe an enclosure or two, connectors to go to outside world, those get kind of project specific.
Here's the last set of stuff I ordered. Some parts I have a specific use in mind, like the 220uF cap for driving AC into a speaker with from a MOSFET amp, 10 pin connector for a JTAG connector, RS232 driver to go with an SPI-driven UART to add 2nd hardware serial port, screw terminlal blocks to connect wires to.
Other parts seemed like something interesting to play with based on things I've seen here - RTC for instance, high current darlington output driver (ULN2803), optoisolators, shunt resistor for measuring current flow.
DIPs are easier to work with, but lately I've been picking up surface mount adapters as well, no special use in mind. Pin spacing on these vary, if you want an adapter board for specific part, make sure they match. I bought a couple of 28 pin adapters, later got 28 pin driver chips, pin spacings happened to match! Just got lucky there. Haven't mated the two yet.
Solder flux is good, isopropyl alchohol and a little brush for cleanup afterwards.
When I used op amps in the past, it was generally for audio purposes, and sometimes for voltage level comparisons to control something else.
For audio, I was buffering inputs to get more gain or to have more current drive capability. Filtering to get allow certain frequency bands thru (low-pass, band-pass, high-pass), mixing signals together, buffering the outputs of D/A converters.
For things like that, read the specs on the op amps and find some that are really low noise, measured in nV/Hz.
http://www.dipmicro.com/store/LF356N These look pretty good - 12nV/Hz.
Read the datasheet on the parts, look at the typical applications, get the support parts they show.
Low power is good also if you are intending to make battery operated things, like a headphone amplifier. I preferred having +/-12 or +/-15 supplies for the ADCs and DACs to avoid DC offset on the AC signals (avoid any distortion added by DC blocking capacitors) and to not have to mess with adding offset to keep signals in the middle of single supply op amps, just one less thing to worry about. Bit of a purist thing I guess, but I had the parts to do that.
This site covers them pretty well.
Some perfboard to build circuits on is handy once you get past the breadboard stage.
Strips of socket pins are very nice to make all your connections to, and then plug in your components. I prefer wirewrap for digital work, not into stripping & soldering lots of little wires, but then my eyes aren't as young as yours.
If you can afford it, a 'scope will come in really handy for op amp circuits. I had this behemoth Tektronix single channel scope that only had 60kHz bandwidth (looking at signals higher than that just dropped right off in amplitude). Later I got a better one.
Now, I use this one www.dpscope.com. Bought it as a kit & assembled, just uses the computer screen for a display. Way handier. $70 plus shipping.
The other thing I have used for looking at audio, picked up via microphone but no reason you couldn't make an adapter to go direct into your sound card, is Visual Analyzer. Free download from http://www.sillanumsoft.org/
Standalone 'arduino' in lower left, LED driver & current limit resistors on upper right, MOSFET amplifier in lower right, 7406 open collector with pullups used as line driver in upper left.
backside showing wirewrap & installation prior to connecting to other parts
Product: 2pin 0.2” / 5mm screw terminal
Product Code: DG126
Product Code: 4N35
Product: LQFP48/TQFP48 Module w/ 228 Proto Pads
Product Code: PCB-LQFP48-DIP48G
Product: 10 Pin Shrouded Male Header
Product Code: IDCM10
Product: DS1307 Dallas Maxim 64 x 8 Serial Real-Time Clock
Product Code: DS1307N
Product: RS232 Transceiver MAX232
Product Code: MAX232
Product: Digital Echo / Surround Processor IC
Product Code: PT2399
Product: Darlington Driver 50V / 500mA ULN2803A ULN2803
Product Code: ULN2803A
Product: 10uF/25V Radial Electrolytic Capacitor
Product Code: C7E10-25-105
Product: 220uF/16V Radial Electrolytic Capacitor
Product Code: C8E22-16.A
Product: Shunt Resistor 10m? 1W ±1% 20ppm
Product Code: PWR4412-2SBR010F
Product: Optocoupler TLP521-1
Product Code: TLP521